[hgtrowvn] reciprocals of odd numbers mod 2^n

odd numbers have multiplicative inverses modulo 2^n (even numbers do not).

? for(n=1, 8, print1(2^n,":"); for(j=1, 2^(n-1), i=2*j-1; print1(" ",lift(Mod(i,2^n)^-1))); print())

for example, in 4th line below, the reciprocals of 1 3 5 7 9 11 13 15 modulo 16 are 1 11 13 7 9 3 5 15 respectively.  the product of each number multiplied by its reciprocal is a number of the form 16N+1.

2: 1

4: 1 3

8: 1 3 5 7

16: 1 11 13 7 9 3 5 15

32: 1 11 13 23 25 3 5 15 17 27 29 7 9 19 21 31

64: 1 43 13 55 57 35 5 47 49 27 61 39 41 19 53 31 33 11 45 23 25 3 37 15 17 59 29 7 9 51 21 63

128: 1 43 77 55 57 35 69 111 113 27 61 39 41 19 53 95 97 11 45 23 25 3 37 79 81 123 29 7 9 115 21 63 65 107 13 119 121 99 5 47 49 91 125 103 105 83 117 31 33 75 109 87 89 67 101 15 17 59 93 71 73 51 85 127

256: 1 171 205 183 57 163 197 239 241 27 61 167 41 19 53 223 225 139 173 151 25 131 165 207 209 251 29 135 9 243 21 191 193 107 141 119 249 99 133 175 177 219 253 103 233 211 245 159 161 75 109 87 217 67 101 143 145 187 221 71 201 179 213 127 129 43 77 55 185 35 69 111 113 155 189 39 169 147 181 95 97 11 45 23 153 3 37 79 81 123 157 7 137 115 149 63 65 235 13 247 121 227 5 47 49 91 125 231 105 83 117 31 33 203 237 215 89 195 229 15 17 59 93 199 73 51 85 255

they look quite random, but there are some patterns, like some early (e.g., 11, 13, 57) and late numbers (e.g., 51, 85) persisting through several exponents.

the randomness is reminiscent of multiplicative inverse modulo a prime number.  however, prime numbers and powers of 2 are about as different as numbers can be.

the reciprocal of a reciprocal is itself, so most numbers pair with another (two-cycle).  however, 1, (2^(n-1) - 1), (2^(n-1) + 1), and (2^n - 1) (the last one equivalent to -1) are always their own inverses (one-cycle).

previously: it is possible to set up a Galois field of order 2^n so that all nonzero elements (not just the odd elements) have reciprocals.  the reduction operation becomes more complicated than integer modulo 2^n.

[sdmzsnne] unobvious composites

a multiplication table between pairs of small prime numbers:

*	13	17	19	23	29	31	37	41	43
13	169	221	247	299	377	403	481	533	559
17	221	289	323	391	493	527	629	697	731
19	247	323	361	437	551	589	703	779	817
23	299	391	437	529	667	713	851	943	989
29	377	493	551	667	841	899	1073	1189	1247
31	403	527	589	713	899	961	1147	1271	1333
37	481	629	703	851	1073	1147	1369	1517	1591
41	533	697	779	943	1189	1271	1517	1681	1763
43	559	731	817	989	1247	1333	1591	1763	1849

divisibility by 2, 3, 5, 11 can be checked by well known rules, and 7 can be checked by short division (requiring effort similar to checks of 3 and 11), so the table starts at 13.  it stops at the last prime (43) whose square is less than the cube of the smallest: (13^3 = 2197) < (47^2 = 2209).

45 table entries, after commutativity.

there are many non-obvious composites (without easy divisibility check) of similar size as entries in the table but not in the table above.  the smallest is 13*47 = 611, so the table contains all non-obvious composites only up to 610.

with that in mind, better is a list rather than a table.  below are the 43 non-obvious composites less than 1000:

169 = 13 * 13
221 = 13 * 17
247 = 13 * 19
289 = 17 * 17
299 = 13 * 23
323 = 17 * 19
361 = 19 * 19
377 = 13 * 29
391 = 17 * 23
403 = 13 * 31
437 = 19 * 23
481 = 13 * 37
493 = 17 * 29
527 = 17 * 31
529 = 23 * 23
533 = 13 * 41
551 = 19 * 29
559 = 13 * 43
589 = 19 * 31
611 = 13 * 47
629 = 17 * 37
667 = 23 * 29
689 = 13 * 53
697 = 17 * 41
703 = 19 * 37
713 = 23 * 31
731 = 17 * 43
767 = 13 * 59
779 = 19 * 41
793 = 13 * 61
799 = 17 * 47
817 = 19 * 43
841 = 29 * 29
851 = 23 * 37
871 = 13 * 67
893 = 19 * 47
899 = 29 * 31
901 = 17 * 53
923 = 13 * 71
943 = 23 * 41
949 = 13 * 73
961 = 31 * 31
989 = 23 * 43

the next entry would be 1003 = 17 * 59.

for divisibility and prime factorization, you only need to know one of the factors, probably the smaller one.

previously, on memorizing the primes less than 1000.  memorizing the composites this way seems competitive and also gets you useful prime factorization.

[wyhgtulu] nearby prime differences

start with a set of consecutive primes.  compute all pairs positive differences among the set, and count the number of times each difference occurs.

note that we are counting differences between primes not necessarily consecutive.  (in contrast, prime gaps, another well studied topic, are differences between consecutive primes.)

running through the differences of all pairs seems like it could be done cleverly faster than quadratic time, but I could not think of a way.  doing all pairs of sets of a million numbers took about half an hour in C++.

among the first 10 odd primes (3 5 7 11 13 17 19 23 29 31), below are the differences from most frequent to least frequent.  each parenthesized record has the format
(difference number-of-occurrences #rank)
where rank may be a range because of ties.  for ease of parsing output by machine, the rank is given as a range even if the range is just one number.

(6 6 #1-1); (2 5 #2-3); (12 5 #2-3); (4 4 #4-6); (8 4 #4-6); (10 4 #4-6); (14 3 #7-9); (16 3 #7-9); (18 3 #7-9); (20 2 #10-12); (24 2 #10-12); (26 2 #10-12); (22 1 #13-14); (28 1 #13-14);

below are the top 100 differences among all pairs of the first 100 odd primes 3 through 547.

(30 60 #1-1); (60 57 #2-2); (42 54 #3-4); (90 54 #3-4); (120 49 #5-5); (6 48 #6-11); (12 48 #6-11); (24 48 #6-11); (36 48 #6-11); (66 48 #6-11); (84 48 #6-11); (18 46 #12-12); (150 45 #13-13); (48 44 #14-16); (126 44 #14-16); (210 44 #14-16); (54 42 #17-18); (78 42 #17-18); (102 40 #19-19); (72 39 #20-21); (180 39 #20-21); (96 38 #22-24); (132 38 #22-24); (168 38 #22-24); (108 37 #25-27); (114 37 #25-27); (156 37 #25-27); (144 35 #28-29); (174 35 #28-29); (138 34 #30-33); (186 34 #30-33); (240 34 #30-33); (270 34 #30-33); (10 33 #34-37); (20 33 #34-37); (70 33 #34-37); (162 33 #34-37); (216 32 #38-38); (40 31 #39-40); (192 31 #39-40); (198 30 #41-45); (204 30 #41-45); (222 30 #41-45); (228 30 #41-45); (252 30 #41-45); (14 29 #46-49); (50 29 #46-49); (56 29 #46-49); (234 29 #46-49); (80 28 #50-53); (140 28 #50-53); (264 28 #50-53); (300 28 #50-53); (4 27 #54-57); (34 27 #54-57); (246 27 #54-57); (330 27 #54-57); (22 26 #58-61); (26 26 #58-61); (44 26 #58-61); (258 26 #58-61); (2 25 #62-66); (28 25 #62-66); (38 25 #62-66); (110 25 #62-66); (276 25 #62-66); (8 24 #67-75); (16 24 #67-75); (32 24 #67-75); (52 24 #67-75); (98 24 #67-75); (100 24 #67-75); (160 24 #67-75); (294 24 #67-75); (360 24 #67-75); (46 23 #76-85); (76 23 #76-85); (82 23 #76-85); (104 23 #76-85); (130 23 #76-85); (154 23 #76-85); (170 23 #76-85); (306 23 #76-85); (312 23 #76-85); (336 23 #76-85); (68 22 #86-91); (74 22 #86-91); (86 22 #86-91); (94 22 #86-91); (112 22 #86-91); (342 22 #86-91); (62 21 #92-98); (64 21 #92-98); (88 21 #92-98); (92 21 #92-98); (190 21 #92-98); (220 21 #92-98); (282 21 #92-98); (116 20 #99-108); (118 20 #99-108); (124 20 #99-108); (146 20 #99-108); (182 20 #99-108); (200 20 #99-108); (230 20 #99-108); (288 20 #99-108); (318 20 #99-108); (390 20 #99-108);

primorials occur commonly among differences.  here are the frequencies of primorial differences among the first 100 primes, extracted from above:

(2 25 #62-66); (6 48 #6-11); (30 60 #1-1); (210 44 #14-16);

I knew in advance that primorials would be common; the motivation for this project was to see what other numbers would be common.  like primorials, the answer turned out to be smooth numbers.  finding the high frequency prime differences feels a little like decoding the music of the primes: these are the strong wavelengths and their overtones.

top differences among the first 1000 odd primes 3 through 7927:

(210 521 #1-1); (420 496 #2-2); (630 486 #3-3); (330 468 #4-4); (840 467 #5-5); (30 454 #6-6); (1050 452 #7-7); (60 447 #8-8); (390 446 #9-9); (1260 445 #10-10); (510 444 #11-12); (660 444 #11-12); (120 442 #13-13); (90 439 #14-14); (150 435 #15-15); (570 433 #16-17); (780 433 #16-17); (180 432 #18-18); (240 431 #19-20); (1470 431 #19-20); (270 430 #21-22); (300 430 #21-22); (990 423 #23-23); (360 421 #24-24); (450 418 #25-26); (690 418 #25-26); (2310 415 #27-27); (480 412 #28-28); (84 410 #29-29); (870 409 #30-31); (1680 409 #30-31); (600 405 #32-32); (42 403 #33-36); (540 403 #33-36); (750 403 #33-36); (1170 403 #33-36); (126 402 #37-37); (930 400 #38-38); (546 399 #39-39); (462 397 #40-40); (1020 394 #41-42); (1890 394 #41-42); (720 390 #43-43); (1320 389 #44-44); (714 388 #45-48); (900 388 #45-48); (960 388 #45-48); (1140 388 #45-48); (336 386 #49-51); (924 386 #49-51); (1560 386 #49-51); (168 385 #52-53); (294 385 #52-53); (66 382 #54-54); (810 380 #55-55); (798 377 #56-57); (1650 377 #56-57); (252 376 #58-59); (1080 376 #58-59); (1230 375 #60-61); (2730 375 #60-61); (504 374 #62-63); (2100 374 #62-63); (1380 371 #64-64); (378 369 #65-67); (1200 369 #65-67); (1530 369 #65-67); (1110 368 #68-68); (1980 366 #69-69); (1092 364 #70-70); (132 361 #71-73); (1386 361 #71-73); (1440 361 #71-73); (672 360 #74-76); (966 360 #74-76); (1290 360 #74-76); (264 358 #77-80); (588 358 #77-80); (858 358 #77-80); (1410 358 #77-80); (204 357 #81-81); (78 356 #82-82); (156 355 #83-84); (396 355 #83-84); (198 354 #85-86); (1350 354 #85-86); (1500 353 #87-91); (1590 353 #87-91); (1710 353 #87-91); (1860 353 #87-91); (2040 353 #87-91); (756 351 #92-92); (882 350 #93-94); (1950 350 #93-94); (234 349 #95-95); (1620 348 #96-97); (2520 348 #96-97); (306 347 #98-98); (102 345 #99-100); (1740 345 #99-100);

primorials 1000:

(2 174 #890-897); (6 343 #103-106); (30 454 #6-6); (210 521 #1-1); (2310 415 #27-27);

among the first 10000 odd primes 3 through 104743:

(2310 4371 #1-1); (4620 4306 #2-2); (2730 4260 #3-3); (5460 4127 #4-4); (3570 4118 #5-5); (6930 4107 #6-6); (3990 4088 #7-7); (210 4058 #8-8); (840 4057 #9-9); (420 4032 #10-11); (9240 4032 #10-11); (630 4027 #12-12); (8190 4011 #13-13); (4830 4000 #14-14); (1890 3993 #15-15); (1470 3992 #16-16); (1260 3980 #17-17); (1050 3977 #18-18); (7140 3969 #19-19); (1680 3964 #20-20); (2100 3949 #21-22); (11550 3949 #21-22); (2940 3923 #23-23); (2520 3917 #24-24); (7980 3908 #25-25); (4290 3902 #26-26); (3150 3897 #27-27); (10920 3886 #28-28); (3780 3878 #29-29); (6090 3876 #30-30); (6510 3873 #31-31); (4200 3866 #32-32); (3360 3865 #33-33); (4410 3855 #34-34); (5040 3832 #35-35); (13860 3820 #36-36); (10710 3815 #37-37); (7770 3800 #38-38); (5250 3795 #39-39); (8610 3774 #40-40); (5670 3769 #41-41); (9660 3760 #42-42); (330 3759 #43-43); (5880 3750 #44-44); (660 3746 #45-45); (9030 3744 #46-46); (6300 3740 #47-48); (13650 3740 #47-48); (11970 3732 #49-49); (6720 3725 #50-50); (5610 3717 #51-51); (1320 3699 #52-53); (1980 3699 #52-53); (1170 3698 #54-54); (8580 3694 #55-55); (7350 3691 #56-56); (990 3689 #57-58); (9870 3689 #57-58); (390 3686 #59-59); (1650 3678 #60-60); (16170 3676 #61-61); (7560 3672 #62-62); (9450 3665 #63-63); (780 3655 #64-64); (1560 3653 #65-65); (6270 3652 #66-66); (14280 3649 #67-67); (16380 3648 #68-68); (6630 3646 #69-69); (8400 3645 #70-70); (12180 3642 #71-71); (8820 3631 #72-72); (2640 3626 #73-73); (2970 3625 #74-74); (11130 3614 #75-75); (18480 3611 #76-76); (7590 3603 #77-77); (3300 3602 #78-78); (1950 3593 #79-79); (3630 3590 #80-80); (3960 3588 #81-81); (10080 3587 #82-82); (14490 3582 #83-83); (570 3569 #84-85); (12390 3569 #84-85); (510 3567 #86-87); (13020 3567 #86-87); (1020 3564 #88-88); (10500 3563 #89-89); (10290 3555 #90-90); (7410 3550 #91-91); (1530 3547 #92-93); (2340 3547 #92-93); (19110 3545 #94-94); (12870 3538 #95-95); (3120 3537 #96-96); (11340 3535 #97-97); (5280 3534 #98-98); (11220 3531 #99-99); (1140 3530 #100-100);

primorials 10000:

(2 1270 #12834-12857); (6 2538 #1642-1643); (30 3449 #127-127); (210 4058 #8-8); (2310 4371 #1-1); (30030 3388 #167-167);

among the first 100000 odd primes 3 through 1299721 (there are no tied ranks):

(30030 38434 #1-1); (60060 37316 #2-2); (39270 37286 #3-3); (43890 36759 #4-4); (90090 36420 #5-5); (2310 36322 #6-6); (46410 36284 #7-7); (4620 36262 #8-8); (53130 36158 #9-9); (6930 36066 #10-10); (51870 36020 #11-11); (9240 35993 #12-12); (11550 35969 #13-13); (78540 35896 #14-14); (13860 35712 #15-15); (18480 35670 #16-16); (16170 35633 #17-17); (20790 35610 #18-18); (2730 35540 #19-19); (23100 35511 #20-20); (5460 35496 #21-21); (87780 35451 #22-22); (25410 35331 #23-23); (27720 35303 #24-24); (8190 35295 #25-25); (120120 35267 #26-26); (10920 35210 #27-27); (66990 35191 #28-28); (71610 35161 #29-29); (32340 35117 #30-30); (13650 35089 #31-31); (16380 35088 #32-32); (36960 35087 #33-33); (19110 35079 #34-34); (62790 35075 #35-35); (34650 35049 #36-36); (41580 34959 #37-37); (3570 34795 #38-38); (92820 34772 #39-39); (21840 34762 #40-40); (117810 34719 #41-41); (46200 34712 #42-42); (32760 34685 #43-43); (7140 34668 #44-44); (24570 34663 #45-45); (48510 34661 #46-46); (27300 34622 #47-47); (50820 34585 #48-48); (10710 34583 #49-49); (67830 34527 #50-50); (150150 34489 #51-51); (3990 34457 #52-52); (55440 34419 #53-53); (7980 34391 #54-54); (38220 34348 #55-55); (17850 34342 #56-56); (35490 34340 #57-57); (79170 34320 #58-58); (57750 34315 #59-59); (85470 34300 #60-60); (14280 34292 #61-61); (106260 34273 #62-62); (40950 34187 #63-63); (43680 34175 #64-64); (62370 34134 #65-65); (103740 34117 #66-66); (11970 34112 #67-67); (21420 34086 #68-68); (64680 34063 #69-69); (84630 34039 #70-70); (4830 34028 #71-71); (19950 34018 #72-72); (15960 34008 #73-73); (69300 33972 #74-74); (94710 33948 #75-75); (99330 33928 #76-76); (24990 33921 #77-77); (9660 33906 #78-78); (131670 33903 #79-79); (49140 33898 #80-80); (73920 33833 #81-81); (28560 33827 #82-82); (23940 33808 #83-83); (54600 33771 #84-84); (32130 33719 #85-85); (82110 33715 #86-86); (14490 33706 #87-87); (27930 33688 #88-88); (76230 33655 #89-89); (6510 33641 #90-90); (31920 33566 #91-91); (80850 33564 #92-92); (83160 33557 #93-93); (6090 33555 #94-94); (65520 33532 #95-95); (35700 33523 #96-96); (57330 33518 #97-97); (19320 33511 #98-98); (12180 33503 #99-99); (24150 33491 #100-100);

primorials 100000:

(2 10250 #158826-158865); (6 20472 #20178-20185); (30 27434 #2003-2003); (210 32719 #161-161); (2310 36322 #6-6); (30030 38434 #1-1); (510510 25233 #5165-5165);

among the first 1000000 (10^6) odd primes 3 through 15485867 (there are no tied ranks):

(510510 340191 #1-1); (570570 335991 #2-2); (30030 331644 #3-3); (60060 330432 #4-4); (120120 329009 #5-5); (690690 328971 #6-6); (90090 328938 #7-7); (150150 327747 #8-8); (180180 327518 #9-9); (1021020 326622 #10-10); (210210 326022 #11-11); (240240 325472 #12-12); (270270 324716 #13-13); (746130 324257 #14-14); (300300 323675 #15-15); (39270 323557 #16-16); (330330 322956 #17-17); (78540 322560 #18-18); (360360 322340 #19-19); (390390 321814 #20-20); (1141140 321719 #21-21); (117810 321414 #22-22); (43890 321349 #23-23); (870870 321329 #24-24); (420420 321016 #25-25); (157080 320339 #26-26); (450450 319941 #27-27); (87780 319841 #28-28); (480480 319488 #29-29); (196350 319001 #30-30); (131670 318880 #31-31); (930930 318868 #32-32); (235620 318821 #33-33); (46410 318065 #34-34); (540540 317717 #35-35); (274890 317417 #36-36); (175560 317340 #37-37); (53130 317215 #38-38); (903210 316926 #39-39); (600600 316875 #40-40); (219450 316635 #41-41); (92820 316210 #42-42); (106260 315973 #43-43); (314160 315965 #44-44); (353430 315806 #45-45); (630630 315502 #46-46); (263340 315340 #47-47); (660660 315117 #48-48); (139230 314979 #49-49); (392700 314930 #50-50); (51870 314922 #51-51); (881790 314887 #52-52); (159390 314726 #53-53); (1531530 314692 #54-54); (185640 314104 #55-55); (307230 313971 #56-56); (66990 313959 #57-57); (103740 313791 #58-58); (1111110 313519 #59-59); (720720 313463 #60-60); (431970 313352 #61-61); (212520 313175 #62-62); (71610 313060 #63-63); (232050 312817 #64-64); (351120 312815 #65-65); (155610 312760 #66-66); (1381380 312727 #67-67); (750750 312487 #68-68); (471240 312399 #69-69); (780780 312344 #70-70); (133980 312153 #71-71); (395010 312101 #72-72); (1009470 312069 #73-73); (62790 311718 #74-74); (810810 311601 #75-75); (265650 311572 #76-76); (143220 311541 #77-77); (278460 311218 #78-78); (549780 311013 #79-79); (207480 310874 #80-80); (85470 310859 #81-81); (840840 310762 #82-82); (438900 310691 #83-83); (318780 310577 #84-84); (324870 310469 #85-85); (200970 310404 #86-86); (482790 310057 #87-87); (589050 309915 #88-88); (94710 309757 #89-89); (125580 309724 #90-90); (259350 309485 #91-91); (371280 309413 #92-92); (1231230 309364 #93-93); (214830 309355 #94-94); (371910 309351 #95-95); (900900 309325 #96-96); (99330 309314 #97-97); (628320 309159 #98-98); (170940 308915 #99-99); (526680 308795 #100-100);

primorials 1000000:

(2 86027 #1894840-1894883); (6 170910 #250608-250620); (30 228548 #26112-26114); (210 274349 #2118-2118); (2310 305046 #144-144); (30030 331644 #3-3); (510510 340191 #1-1); (9699690 141233 #642368-642376);

among the first 10^6 primes after 10^10, namely from 10^10+19 through 10^10+23010139 (there are no tied ranks):

(510510 232043 #1-1); (570570 230166 #2-2); (1021020 226921 #3-3); (690690 225737 #4-4); (1141140 224221 #5-5); (746130 222868 #6-6); (60060 221915 #7-7); (870870 221831 #8-8); (30030 221797 #9-9); (150150 221644 #10-10); (1531530 221466 #11-11); (90090 221355 #12-12); (210210 221315 #13-13); (120120 221255 #14-14); (930930 220820 #15-15); (180180 220629 #16-16); (270270 220193 #17-17); (300300 219734 #18-18); (240240 219460 #19-19); (1381380 219454 #20-20); (330330 219211 #21-21); (360360 218845 #22-22); (903210 218765 #23-23); (390390 218595 #24-24); (420420 218552 #25-25); (881790 218101 #26-26); (480480 218059 #27-27); (1111110 217837 #28-28); (450450 217779 #29-29); (1711710 217739 #30-30); (39270 217635 #31-31); (540540 217338 #32-32); (2042040 217082 #33-33); (600600 216622 #34-34); (630630 216617 #35-35); (78540 216501 #36-36); (117810 216404 #37-37); (1009470 216351 #38-38); (1231230 216241 #39-39); (720720 216045 #40-40); (196350 215994 #41-41); (157080 215829 #42-42); (43890 215797 #43-43); (660660 215533 #44-44); (235620 215460 #45-45); (1492260 215392 #46-46); (1291290 215335 #47-47); (750750 215144 #48-48); (780780 214788 #49-49); (274890 214773 #50-50); (810810 214767 #51-51); (131670 214740 #52-52); (87780 214737 #53-53); (314160 214733 #54-54); (840840 214580 #55-55); (353430 214368 #56-56); (1138830 214271 #57-57); (900900 214269 #58-58); (175560 213720 #59-59); (219450 213694 #60-60); (1741740 213676 #61-61); (46410 213530 #62-62); (392700 213441 #63-63); (307230 213434 #64-64); (1217370 213430 #65-65); (1067430 213417 #66-66); (431970 213342 #67-67); (263340 213308 #68-68); (960960 213186 #69-69); (92820 213075 #70-70); (471240 213069 #71-71); (1411410 213067 #72-72); (351120 212893 #73-73); (53130 212760 #74-74); (589050 212729 #75-75); (990990 212645 #76-76); (2072070 212598 #77-77); (159390 212486 #78-78); (139230 212460 #79-79); (106260 212415 #80-80); (1051050 212366 #81-81); (395010 212241 #82-82); (549780 212217 #83-83); (1272810 212202 #84-84); (2282280 212191 #85-85); (1081080 212140 #86-86); (185640 212057 #87-87); (1861860 211670 #88-88); (51870 211619 #89-89); (1591590 211566 #90-90); (628320 211537 #91-91); (265650 211535 #92-92); (212520 211497 #93-93); (232050 211469 #94-94); (324870 211438 #95-95); (1171170 211347 #96-96); (438900 211271 #97-97); (482790 211263 #98-98); (278460 211262 #99-99); (1201200 211197 #100-100);

primorials at 10^10:

(2 57382 #2903318-2903469); (6 114833 #392781-392814); (30 152755 #45239-45242); (210 183875 #3655-3655); (2310 204068 #307-307); (30030 221797 #9-9); (510510 232043 #1-1); (9699690 145066 #85780-85787);

among the first 10^6 primes after 10^100, namely 10^100+267 through 10^100+230441007:

(9699690 24034 #1-1); (11741730 23765 #2-2); (1531530 23622 #3-3); (1021020 23590 #4-4); (2282280 23564 #5-5); (3063060 23542 #6-6); (2552550 23519 #7-7); (510510 23497 #8-8); (2042040 23482 #9-9); (1141140 23475 #10-10); (570570 23462 #11-11); (1711710 23418 #12-12); (3423420 23406 #13-13); (690690 23388 #14-14); (3573570 23324 #15-15); (4594590 23270 #16-16); (4564560 23239 #17-18); (13123110 23239 #17-18); (4084080 23237 #19-19); (1381380 23115 #20-21); (5705700 23115 #20-21); (3993990 23109 #22-22); (6276270 23090 #23-23); (2852850 23083 #24-24); (14804790 23037 #25-25); (4144140 23006 #26-26); (870870 22993 #27-27); (6636630 22968 #28-28); (7147140 22960 #29-29); (6126120 22948 #30-30); (8678670 22946 #31-31); (5135130 22937 #32-32); (7417410 22929 #33-33); (2072070 22922 #34-34); (5615610 22916 #35-35); (19399380 22908 #36-36); (15825810 22900 #37-37); (6846840 22887 #38-38); (2984520 22881 #39-39); (746130 22880 #40-40); (7987980 22858 #41-41); (7657650 22852 #42-42); (3730650 22843 #43-43); (2238390 22838 #44-44); (6216210 22837 #45-45); (3453450 22835 #46-46); (1741740 22821 #47-47); (9189180 22820 #48-48); (4834830 22799 #49-49); (1111110 22790 #50-50); (930930 22756 #51-51); (2762760 22747 #52-52); (5105100 22735 #53-54); (8168160 22735 #53-54); (5222910 22732 #55-55); (3723720 22713 #56-56); (1411410 22709 #57-57); (5969040 22708 #58-58); (2792790 22703 #59-59); (10840830 22702 #60-60); (2612610 22696 #61-61); (1231230 22693 #62-62); (6906900 22674 #63-63); (3333330 22672 #64-64); (903210 22667 #65-65); (1492260 22640 #66-66); (1591590 22625 #67-67); (8288280 22624 #68-68); (2222220 22623 #69-69); (4444440 22619 #70-70); (16546530 22607 #71-71); (9129120 22587 #72-72); (4354350 22586 #73-74); (5585580 22586 #73-74); (3483480 22577 #75-75); (1861860 22575 #76-76); (10720710 22566 #77-77); (5225220 22565 #78-78); (2645370 22551 #79-79); (7597590 22542 #80-80); (8558550 22538 #81-81); (5525520 22537 #82-82); (12252240 22534 #83-83); (10210200 22526 #84-84); (60060 22512 #85-86); (11231220 22512 #85-86); (881790 22510 #87-87); (12762750 22500 #88-88); (1009470 22485 #89-89); (2132130 22482 #90-90); (13783770 22472 #91-91); (1291290 22468 #92-92); (1806420 22457 #93-94); (2709630 22457 #93-94); (2277660 22456 #95-95); (2582580 22448 #96-96); (2822820 22447 #97-97); (4654650 22446 #98-98); (2492490 22435 #99-99); (11981970 22425 #100-100);

primorials at 10^100:

(2 5767 #28706071-28717948); (6 11484 #3922267-3924636); (30 15466 #384556-384851); (210 18481 #32555-32582); (2310 20373 #3393-3398); (30030 22276 #135-135); (510510 23497 #8-8); (9699690 24034 #1-1); (223092870 815 #102930136-102945303);

future work: 10^1000.

[ahlvcwcw] factoring while you wait

on a modern computer, Pari/GP can usually factor RSA-style numbers (the worst case for factoring) of 60 digits (199 bits) in 30 seconds or less, subjectively the limit for human interactive patience.  factoring algorithms are probabilistic (randomized), so there is no hard upper bound to "usually".

74 digits (246 bits) usually takes under 5 minutes (300 seconds).  with improvements in software and parallel processing (SMP) (Pari/GP's ECM and MPQS are single-threaded), one could imagine getting getting 74-digits down to under 30 seconds.

previously, standard deviation.

[vyakuwax] ordered dice

roll some dice, order them by some metric independent of what number the die shows, then read off the numbers in that order.

example dice ordering metrics: color (ordering by rainbow), size, number of sides.  the last one provides a mixed-radix random number.  previously, dice with two digits on each face.

rolling a whole bunch of dice all at once is more efficient (and fun) than rolling one die repeatedly.  (more efficient until the (log n) factor of sorting dominates.)  dice collisions induce more randomness.

ordered d2: coins by value, size, or year.

ordered 30 * d20 generates 128 bits, assuming they are uniform, which is not a good assumption: d20s are likely not manufactured to high precision as casino dice are.  ordered 26d20 generates 112 bits (Triple DES): give each d20 an identifier letter.

or, a funnel which forces them single file.  cubes might be best because they pack.  funnel by itself is probably not enough.

(then, feed the sequence of random numbers into a hash function, or seed a cryptographically secure pseudorandom number generator with them.  even if the dice are a little biased, the output will be pretty random.)

[egfgwujd] twin primes adjacent to highly composite numbers

below are some numbers N with many divisors and for which N-1 and N+1 are both prime, i.e., twin primes.

the suffix A indicates a highly composite number (sets a new record for number of divisors among all numbers, not necessarily those between twin primes); B similarly indicates it ties the record for number of divisors (largely composite number); C indicates a new second place in number of divisors; D ties second place number of divisors.

4/A 6/A 12/A 18/A 30/B 42/D 60/A 72/B 108/B 180/A 240/A 420/B 432/D 600/A 660/B 1320/D 2340/D 3360/A 5280/D 5880/D 6300/C 7560/A 9240/A 21840/D 35280/D 42840/B 55440/A 65520/A 92400/D 100800/C 110880/A 128520/B 180180/D 415800/B 453600/D 514080/D 526680/D 540540/D 1867320/D 1912680/D 1921920/D 1940400/C 2489760/D 6652800/D 6846840/D 7068600/D 28828800/B 31600800/B 34594560/D 85765680/D 100900800/C 121080960/D 232792560/A 287567280/D 397837440/D 634888800/D 845404560/D 1259818560/D 1470268800/B 1574773200/D 6299092800/D 10708457760/D 12681068400/B 23827003200/D 32125373280/B 32590958400/C 33816182400/A 34918884000/B 40156716600/B 73329656400/A 135019684800/D 216497080800/B 439977938400/D 449755225920/B 578256719040/D 2126560035600/D 2835413380800/B 2971597028400/D 6278415343200/B 18632716502400/A 18835246029600/C 19275223968000/B 59753194300800/B 62403537596400/D 69712060017600/B 92199821313600/B 130429015516800/A 279490747536000/C 313704270079200/B 448148957256000/D 838472242608000/D 1382997319704000/D 1402111916805600/B 1498809290378400/D

investigated 1169 numbers from 1 to 1516237305382800 in those categories (A, B, C, D) previously generated.  95 numbers (above) are surrounded by twin primes.

inspired by some numbers between twin primes being quite smooth.  if searching for large twin primes (not sure why one would want to do this), is it profitable to restrict to those bracketing a smooth number?  (seems yes.)  if so, how smooth?  perhaps Pierpont, or some other sequence of very smooth numbers.

[dwrhiuki] timing factorization of random 200-bit numbers

200 bit: mean= 534 ms, std= 1062 ms, n= 10000
201 bit: mean= 564 ms, std= 1112 ms, n= 10000
202 bit: mean= 599 ms, std= 1186 ms, n= 10000

all are about 61 digits (future post ahlvcwcw).

Chebyshev's inequality (because the distribution of factoring times is most definitely not a normal distribution) says 99% are within 10 standard deviations of the mean.  (can we do better than Chebyshev knowing that all values must be positive?)

to get standard error of the mean, divide standard deviation by sqrt(n) = 100.

model name : Intel(R) Core(TM) i7-4810MQ CPU @ 2.80GHz

GP/PARI CALCULATOR Version 2.15.4 (released)
amd64 running linux (x86-64/GMP-6.2.1 kernel) 64-bit version
compiled: Jul 12 2023, gcc version 12.3.0 (Debian 12.3.0-5)
threading engine: pthread

? for(bitsize=200, 202, gettime; sumt=0; s2t=0; for(i=1, 10000, s=random(2^bitsize); print("dataline ",bitsize," ",s," ",factor(s)); t=gettime; sumt+=t; s2t+=t*t; print("i ",i," t ",t," sumt ",sumt," s2t ",s2t)); print("done ",bitsize))

[djtfvrnt] sigil numbers

IEEE 754 floating point can represent infinities and Not a Number without crashing (in contrast to integers, which cannot represent those):

float infinity = 1/0.0;
float NaN = 0/0.0;
float negative_infinity = -1/0.0;

the denominator of 0.0 not 0 is needed to avoid a compile-time warning (gcc 7.5.0). 1.0/0 and 0.0/0 also cause the warning.

the infinities may be useful for algorithms which require an initialization value larger or smaller than any value that will be seen in the input, for example, finding the maximum of an array.

I can't think of any numerical use case for NaN.  there are apparently many encodings (bit strings) that are NaN, so maybe useful for steganography.

[bvqqdwkp] Pierpont twin primes

below are the 73 twin primes of the form (2^a * 3^b plus and minus 1) less than 2^10000 ~= 10^3010.

(2^2 * 3^0 +- 1), (2^1 * 3^1 +- 1), (2^2 * 3^1 +- 1), (2^1 * 3^2 +- 1), (2^3 * 3^2 +- 1), (2^2 * 3^3 +- 1), (2^6 * 3^1 +- 1), (2^4 * 3^3 +- 1), (2^7 * 3^2 +- 1), (2^5 * 3^4 +- 1), (2^6 * 3^7 +- 1), (2^3 * 3^10 +- 1), (2^18 * 3^1 +- 1), (2^12 * 3^5 +- 1), (2^2 * 3^15 +- 1), (2^18 * 3^5 +- 1), (2^21 * 3^4 +- 1), (2^24 * 3^5 +- 1), (2^27 * 3^4 +- 1), (2^30 * 3^7 +- 1), (2^33 * 3^8 +- 1), (2^43 * 3^2 +- 1), (2^32 * 3^9 +- 1), (2^36 * 3^7 +- 1), (2^11 * 3^24 +- 1), (2^31 * 3^12 +- 1), (2^43 * 3^8 +- 1), (2^32 * 3^15 +- 1), (2^50 * 3^9 +- 1), (2^63 * 3^2 +- 1), (2^66 * 3^25 +- 1), (2^79 * 3^20 +- 1), (2^99 * 3^10 +- 1), (2^57 * 3^64 +- 1), (2^82 * 3^63 +- 1), (2^148 * 3^27 +- 1), (2^63 * 3^88 +- 1), (2^56 * 3^99 +- 1), (2^211 * 3^2 +- 1), (2^275 * 3^16 +- 1), (2^287 * 3^10 +- 1), (2^90 * 3^169 +- 1), (2^148 * 3^135 +- 1), (2^298 * 3^51 +- 1), (2^160 * 3^141 +- 1), (2^363 * 3^52 +- 1), (2^134 * 3^231 +- 1), (2^49 * 3^320 +- 1), (2^529 * 3^44 +- 1), (2^264 * 3^419 +- 1), (2^960 * 3^143 +- 1), (2^541 * 3^476 +- 1), (2^988 * 3^207 +- 1), (2^1015 * 3^332 +- 1), (2^1440 * 3^97 +- 1), (2^1295 * 3^324 +- 1), (2^979 * 3^738 +- 1), (2^258 * 3^1493 +- 1), (2^637 * 3^1320 +- 1), (2^2320 * 3^333 +- 1), (2^1036 * 3^1167 +- 1), (2^2815 * 3^188 +- 1), (2^1063 * 3^1440 +- 1), (2^180 * 3^2251 +- 1), (2^888 * 3^2033 +- 1), (2^300 * 3^2819 +- 1), (2^2176 * 3^2175 +- 1), (2^4014 * 3^1879 +- 1), (2^280 * 3^4311 +- 1), (2^6228 * 3^571 +- 1), (2^6981 * 3^560 +- 1), (2^3505 * 3^2892 +- 1), (2^5899 * 3^2570 +- 1)

is this a statistically high yield of twin primes?

also previously.

[mawvhdtz] permutation by primes

we permute the numbers 1..n using the randomness of the prime numbers.

there are many ways to do this.  we implement the "inside-out" variation of the Fisher-Yates shuffle.  the index "c" chains iterations of the loop, because it feels like that would make things more "random". we transform primes by p=(p-1)/2 to prevent them from being all odd.  because the i-th prime is approximately i*log(i), c will cycle around (because of mod) about log(i)/2 times each iteration.

u(n) = my(a=vector(n)); my(c=0); for(i=1, n, my(p=prime(i)); if(2==p, p=0, p=(p-1)/2); c+=p; c%=i; j=c+1; a[i]=a[j]; a[j]=i); a;

note that vectors in Pari/GP are 1-indexed and initialized to zero.  we are permuting numbers 1 through n, so if we were to see a zero in the output, an uninitialized value has gotten through, i.e., there is a bug.  the nthprime function is also 1-indexed, i.e., prime(1)=2 .

here is the permutation of numbers 1 through 30:

u(30)

[3, 10, 29, 6, 18, 15, 28, 21, 5, 27, 17, 12, 26, 20, 25, 16, 24, 7, 19, 11, 23, 1, 9, 8, 13, 14, 2, 4, 30, 22]

step by step: the newest number each iteration is underlined.

i	prime(i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
1	2	1
2	3	1	2
3	5	3	2	1
4	7	3	2	1	4
5	11	3	2	1	5	4
6	13	3	2	1	6	4	5
7	17	3	2	1	6	7	5	4
8	19	3	2	1	6	7	8	4	5
9	23	3	2	1	6	7	8	4	9	5
10	29	3	10	1	6	7	8	4	9	5	2
11	31	3	10	1	6	7	11	4	9	5	2	8
12	37	3	10	1	6	7	11	4	9	5	2	8	12
13	41	3	10	1	6	7	13	4	9	5	2	8	12	11
14	43	3	10	1	6	7	13	4	9	5	2	8	12	14	11
15	47	3	10	1	6	7	15	4	9	5	2	8	12	14	11	13
16	53	3	10	1	6	7	15	4	9	5	2	8	12	14	11	13	16
17	59	3	10	1	6	7	15	4	9	5	2	17	12	14	11	13	16	8
18	61	3	10	1	6	18	15	4	9	5	2	17	12	14	11	13	16	8	7
19	67	3	10	1	6	18	15	4	9	5	2	17	12	14	11	13	16	8	7	19
20	71	3	10	1	6	18	15	4	9	5	2	17	12	14	20	13	16	8	7	19	11
21	73	3	10	1	6	18	15	4	21	5	2	17	12	14	20	13	16	8	7	19	11	9
22	79	3	10	22	6	18	15	4	21	5	2	17	12	14	20	13	16	8	7	19	11	9	1
23	83	3	10	22	6	18	15	4	21	5	2	17	12	14	20	13	16	8	7	19	11	23	1	9
24	89	3	10	22	6	18	15	4	21	5	2	17	12	14	20	13	16	24	7	19	11	23	1	9	8
25	97	3	10	22	6	18	15	4	21	5	2	17	12	14	20	25	16	24	7	19	11	23	1	9	8	13
26	101	3	10	22	6	18	15	4	21	5	2	17	12	26	20	25	16	24	7	19	11	23	1	9	8	13	14
27	103	3	10	22	6	18	15	4	21	5	27	17	12	26	20	25	16	24	7	19	11	23	1	9	8	13	14	2
28	107	3	10	22	6	18	15	28	21	5	27	17	12	26	20	25	16	24	7	19	11	23	1	9	8	13	14	2	4
29	109	3	10	29	6	18	15	28	21	5	27	17	12	26	20	25	16	24	7	19	11	23	1	9	8	13	14	2	4	22
30	113	3	10	29	6	18	15	28	21	5	27	17	12	26	20	25	16	24	7	19	11	23	1	9	8	13	14	2	4	30	22
i	prime(i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30

note that the inside-out algorithm can be run incrementally, so terminating the algorithm at any step above would result in a permutation of a fewer number of elements.

permuting the digits 0 through 9 yields 2905673841.  permuting the letters a through z yields cjvfroduebqlztypxgskwaihmn.

motivation is a puzzle that requires solving the Riemann Hypothesis.  consider using the above permutation (and additional permutations generated similarly) in a substitution-permutation cipher.

how random are these permutations?  although we've corrected for the bias against even numbers, primes not being multiples of each other likely introduces other biases.  what other structures that look approximately uniformly random can be constructed from the primes?  (previously, bitmaps of gradually decreasing density.)  if we first produce a good cipher, then anything is possible.

[untgildg] second collision resistance

collisions in hash functions are unavoidable.  if you find one collision (messages A and B have the same hash), how easy is it to manufacture another collision (messages C and D have the same hash), or even an infinite collection of collisions?

can cryptographic hash functions be designed to prevent collisions from easily yielding more collisions?

[vsibrkyz] astronomically rare dice rolls

obtaining a whole bunch of regular dice (d6) and rolling them simultaneously seems not very difficult.  (Amazon currently sells sets of as many as 100 for under $20.)  consider rolling until they all come up 6.  doing such an exercise allows viscerally trying to experience very low probabilities.  humans generally have difficulty understanding low probabilities.  for a given low probability, you can map it to a number of dice.

below are the reciprocals of probabilities of N dice up to 100, interspersed with some orders of magnitude in base 10 and base 2.  for example, all sixes on 2 dice has a probability of 1 in 36.  100 dice is approximately the probability of breaking 258-bit security by random chance.

is it practically difficult to roll many dice simultaneously?  maybe they land leaned against each other, interfering with landing flat.  beyond a certain number, it becomes difficult to verify between rolls that you haven't lost any.

coins are also easy to obtain, but when flipping many coins simultaneously, it's likely some will roll quite far on their edges before landing on a side.  collisions among dice also might occasionally propel some of them a very long distance.

1 6 (6.0e0)
2 36 (3.6e1)
3 216 (2.2e2)
4 1,296 (1.3e3)
5 7,776 (7.8e3)
6 46,656 (4.7e4)
7 279,936 (2.8e5)
million
8 1,679,616 (1.7e6)
9 10,077,696 (1.0e7)
10 60,466,176 (6.0e7)
11 362,797,056 (3.6e8)
billion
12 2,176,782,336 (2.2e9)
13 13,060,694,016 (1.3e10)
14 78,364,164,096 (7.8e10)
15 470,184,984,576 (4.7e11)
trillion
16 2,821,109,907,456 (2.8e12)
17 16,926,659,444,736 (1.7e13)
18 101,559,956,668,416 (1.0e14)
19 609,359,740,010,496 (6.1e14)
quadrillion
20 3,656,158,440,062,976 (3.7e15)
21 21,936,950,640,377,856 (2.2e16)
22 131,621,703,842,267,136 (1.3e17)
23 789,730,223,053,602,816 (7.9e17)
24 4,738,381,338,321,616,896 (4.7e18)
64 bits
25 28,430,288,029,929,701,376 (2.8e19)
26 170,581,728,179,578,208,256 (1.7e20)
27 1,023,490,369,077,469,249,536 (1.0e21)
28 6,140,942,214,464,815,497,216 (6.1e21)
29 36,845,653,286,788,892,983,296 (3.7e22)
30 221,073,919,720,733,357,899,776 (2.2e23)
31 1,326,443,518,324,400,147,398,656 (1.3e24)
32 7,958,661,109,946,400,884,391,936 (8.0e24)
33 47,751,966,659,678,405,306,351,616 (4.8e25)
34 286,511,799,958,070,431,838,109,696 (2.9e26)
35 1,719,070,799,748,422,591,028,658,176 (1.7e27)
36 10,314,424,798,490,535,546,171,949,056 (1.0e28)
37 61,886,548,790,943,213,277,031,694,336 (6.2e28)
38 371,319,292,745,659,279,662,190,166,016 (3.7e29)
39 2,227,915,756,473,955,677,973,140,996,096 (2.2e30)
40 13,367,494,538,843,734,067,838,845,976,576 (1.3e31)
41 80,204,967,233,062,404,407,033,075,859,456 (8.0e31)
42 481,229,803,398,374,426,442,198,455,156,736 (4.8e32)
43 2,887,378,820,390,246,558,653,190,730,940,416 (2.9e33)
44 17,324,272,922,341,479,351,919,144,385,642,496 (1.7e34)
45 103,945,637,534,048,876,111,514,866,313,854,976 (1.0e35)
46 623,673,825,204,293,256,669,089,197,883,129,856 (6.2e35)
47 3,742,042,951,225,759,540,014,535,187,298,779,136 (3.7e36)
48 22,452,257,707,354,557,240,087,211,123,792,674,816 (2.2e37)
49 134,713,546,244,127,343,440,523,266,742,756,048,896 (1.3e38)
128 bits
50 808,281,277,464,764,060,643,139,600,456,536,293,376 (8.1e38)
51 4,849,687,664,788,584,363,858,837,602,739,217,760,256 (4.8e39)
52 29,098,125,988,731,506,183,153,025,616,435,306,561,536 (2.9e40)
53 174,588,755,932,389,037,098,918,153,698,611,839,369,216 (1.7e41)
54 1,047,532,535,594,334,222,593,508,922,191,671,036,215,296 (1.0e42)
55 6,285,195,213,566,005,335,561,053,533,150,026,217,291,776 (6.3e42)
56 37,711,171,281,396,032,013,366,321,198,900,157,303,750,656 (3.8e43)
57 226,267,027,688,376,192,080,197,927,193,400,943,822,503,936 (2.3e44)
58 1,357,602,166,130,257,152,481,187,563,160,405,662,935,023,616 (1.4e45)
59 8,145,612,996,781,542,914,887,125,378,962,433,977,610,141,696 (8.1e45)
60 48,873,677,980,689,257,489,322,752,273,774,603,865,660,850,176 (4.9e46)
61 293,242,067,884,135,544,935,936,513,642,647,623,193,965,101,056 (2.9e47)
62 1,759,452,407,304,813,269,615,619,081,855,885,739,163,790,606,336 (1.8e48)
63 10,556,714,443,828,879,617,693,714,491,135,314,434,982,743,638,016 (1.1e49)
64 63,340,286,662,973,277,706,162,286,946,811,886,609,896,461,828,096 (6.3e49)
65 380,041,719,977,839,666,236,973,721,680,871,319,659,378,770,968,576 (3.8e50)
66 2,280,250,319,867,037,997,421,842,330,085,227,917,956,272,625,811,456 (2.3e51)
67 13,681,501,919,202,227,984,531,053,980,511,367,507,737,635,754,868,736 (1.4e52)
68 82,089,011,515,213,367,907,186,323,883,068,205,046,425,814,529,212,416 (8.2e52)
69 492,534,069,091,280,207,443,117,943,298,409,230,278,554,887,175,274,496 (4.9e53)
70 2,955,204,414,547,681,244,658,707,659,790,455,381,671,329,323,051,646,976 (3.0e54)
71 17,731,226,487,286,087,467,952,245,958,742,732,290,027,975,938,309,881,856 (1.8e55)
72 106,387,358,923,716,524,807,713,475,752,456,393,740,167,855,629,859,291,136 (1.1e56)
73 638,324,153,542,299,148,846,280,854,514,738,362,441,007,133,779,155,746,816 (6.4e56)
74 3,829,944,921,253,794,893,077,685,127,088,430,174,646,042,802,674,934,480,896 (3.8e57)
192 bits
75 22,979,669,527,522,769,358,466,110,762,530,581,047,876,256,816,049,606,885,376 (2.3e58)
76 137,878,017,165,136,616,150,796,664,575,183,486,287,257,540,896,297,641,312,256 (1.4e59)
77 827,268,102,990,819,696,904,779,987,451,100,917,723,545,245,377,785,847,873,536 (8.3e59)
78 4,963,608,617,944,918,181,428,679,924,706,605,506,341,271,472,266,715,087,241,216 (5.0e60)
79 29,781,651,707,669,509,088,572,079,548,239,633,038,047,628,833,600,290,523,447,296 (3.0e61)
80 178,689,910,246,017,054,531,432,477,289,437,798,228,285,773,001,601,743,140,683,776 (1.8e62)
81 1,072,139,461,476,102,327,188,594,863,736,626,789,369,714,638,009,610,458,844,102,656 (1.1e63)
82 6,432,836,768,856,613,963,131,569,182,419,760,736,218,287,828,057,662,753,064,615,936 (6.4e63)
83 38,597,020,613,139,683,778,789,415,094,518,564,417,309,726,968,345,976,518,387,695,616 (3.9e64)
84 231,582,123,678,838,102,672,736,490,567,111,386,503,858,361,810,075,859,110,326,173,696 (2.3e65)
85 1,389,492,742,073,028,616,036,418,943,402,668,319,023,150,170,860,455,154,661,957,042,176 (1.4e66)
86 8,336,956,452,438,171,696,218,513,660,416,009,914,138,901,025,162,730,927,971,742,253,056 (8.3e66)
87 50,021,738,714,629,030,177,311,081,962,496,059,484,833,406,150,976,385,567,830,453,518,336 (5.0e67)
88 300,130,432,287,774,181,063,866,491,774,976,356,909,000,436,905,858,313,406,982,721,110,016 (3.0e68)
89 1,800,782,593,726,645,086,383,198,950,649,858,141,454,002,621,435,149,880,441,896,326,660,096 (1.8e69)
90 10,804,695,562,359,870,518,299,193,703,899,148,848,724,015,728,610,899,282,651,377,959,960,576 (1.1e70)
91 64,828,173,374,159,223,109,795,162,223,394,893,092,344,094,371,665,395,695,908,267,759,763,456 (6.5e70)
92 388,969,040,244,955,338,658,770,973,340,369,358,554,064,566,229,992,374,175,449,606,558,580,736 (3.9e71)
93 2,333,814,241,469,732,031,952,625,840,042,216,151,324,387,397,379,954,245,052,697,639,351,484,416 (2.3e72)
94 14,002,885,448,818,392,191,715,755,040,253,296,907,946,324,384,279,725,470,316,185,836,108,906,496 (1.4e73)
95 84,017,312,692,910,353,150,294,530,241,519,781,447,677,946,305,678,352,821,897,115,016,653,438,976 (8.4e73)
96 504,103,876,157,462,118,901,767,181,449,118,688,686,067,677,834,070,116,931,382,690,099,920,633,856 (5.0e74)
97 3,024,623,256,944,772,713,410,603,088,694,712,132,116,406,067,004,420,701,588,296,140,599,523,803,136 (3.0e75)
98 18,147,739,541,668,636,280,463,618,532,168,272,792,698,436,402,026,524,209,529,776,843,597,142,818,816 (1.8e76)
99 108,886,437,250,011,817,682,781,711,193,009,636,756,190,618,412,159,145,257,178,661,061,582,856,912,896 (1.1e77)
256 bits
100 653,318,623,500,070,906,096,690,267,158,057,820,537,143,710,472,954,871,543,071,966,369,497,141,477,376 (6.5e77)

[aoypoxfn] factorization challenges easy to type

each composite number below consists of repetitions of the digit string 1234567890 except the final few digits.  the smaller factors are large enough that quadratic sieve is probably more efficient than elliptic curve method (ECM).

previously similar for base 2, assuming you have an expression parser.

40 digits: 1234567890123456789012345678901234567759 = 5284744045008013 * 233609779321220631386443

50 digits: 12345678901234567890123456789012345678901234567633 = 92279162895906968524229 * 133786203881810018387538077

60 digits: 123456789012345678901234567890123456789012345678901234567831 = 87988560978576315942371700757 * 1403100444413510291678340665083

70 digits: 1234567890123456789012345678901234567890123456789012345678901234566713 = 61208527479308458227554760703 * 20169867516267279020392646921656369545671

80 digits: 12345678901234567890123456789012345678901234567890123456789012345678901234567751 = 575099128882847693294536030177708942207 * 21467045038333698701383939676398791777593

90 digits: 123456789012345678901234567890123456789012345678901234567890123456789012345678901234567597 = 40166540925345636983244707230202848993 * 3073622626399545288925403158158369054873454157207629

[hvrryxye] trial division before ispseudoprime in Pari/GP

below is Pari/GP code which prints out a table for the optimal amount of trial division to do before calling ispseudoprime on large primes, automating some of previously discussed.  the table is specific to the computer on which the code is run, specific to the relative speeds of trial division and modular exponentiation.

ispseudoprime internally does trial division only up to 101, so parithreshold is 102.  to confirm this, we use Mersenne prime 2^19937-1.  the first call to ispseudoprome returns immediately.

? #
timer = 1 (on)

? y=2^19937-1;
? ispseudoprime(101*y)
0

? ispseudoprime(103*y)
cpu time = 1,313 ms, real time = 1,316 ms.
0

we construct a composite which will pass ispseudoprime's internal trial division by multiplying large random primes (primes approximately 2^1000 = growthrate).  (unfixed bug: asking for a random prime less than 2^1000 might, with miniscule probability, generate a prime less than 102.)

we do some fine adjustment of growthrate so that the composites are approximately 1000n+buffer bits (buffer = 5) to make the table pretty.  calling randomprime with argument 2^(x+1) will, on average, provide a prime of size 2^x.

the variable "dummy" is to make sure the calls to mod and ispseudoprime do not get optimized away.  this might be more cautious than necessary.

after its small amount of trial division, ispseudoprime does the BPSW primality test, which starts with a Miller-Rabin test with base 2.  when given a composite, with high probability this first test fails, and we assume that this is always the case.  if we are unlucky and hit a composite which has to do the rest of BPSW, then that line's entry in the second column will be too large by a factor of 3 or 4.

the output is probabilistic; do multiple runs to assure yourself of results.

the columns are (1) width in bits of the composite being tested, (2) milliseconds for a failing ispseudoprime test, (3) milliseconds for one trial division, (4) how far (with forprime) you should do trial division for numbers of that size.  for example, if you are testing primality of 10000-bit numbers on the same computer as tested below, you should define a function like this:

mypseudoprime(n) = forprime(p=100, 301720, if(n%p==0,return(0))); ispseudoprime(n)

future work: fit output to a curve; interpolate and extrapolate.

the code:

? print("(bits) (miller-rabin ms) (trial division ms) (forprime limit)"); buffer=5; growthrate=1000; trialdivisioniterations=500000; parithreshold=102; endprime= prime(primepi(parithreshold)+trialdivisioniterations); millerrabiniterations=10; k=randomprime(2^(growthrate+buffer+1)); for(i=2,+oo,desired=growthrate*i+buffer; thisgrow=desired-round(log(k)/log(2)); k*=randomprime(2^(thisgrow+1)); dummy=0; gettime; for(j=1,millerrabiniterations,dummy+=ispseudoprime(k)); timemillerrabin=gettime/millerrabiniterations; forprime(p=parithreshold,endprime,if(k%p==0,dummy+=1)); timetrialdivision=gettime/trialdivisioniterations; printf("%.1f %.1f (%.2e) %.1f\n", log(k)/log(2), timemillerrabin, timetrialdivision, timemillerrabin/timetrialdivision))

example output:

(bits) (miller-rabin ms) (trial division ms) (forprime limit)
2004.2 4.1 (3.12 e-4) 13141.0
3005.2 13.2 (3.84 e-4) 34375.0
4004.5 25.1 (4.50 e-4) 55777.8
5005.4 44.7 (5.20 e-4) 85961.5
6005.5 73.2 (5.92 e-4) 123648.6
7005.7 110.6 (6.60 e-4) 167575.8
8005.7 149.9 (7.26 e-4) 206473.8
8995.9 199.4 (8.00 e-4) 249250.0
10005.2 263.1 (8.72 e-4) 301720.2
11003.1 335.0 (9.40 e-4) 356383.0
12000.7 420.4 (1.01 e-3) 417892.6
13005.2 519.3 (1.07 e-3) 484421.6
14003.4 612.3 (1.14 e-3) 535227.3
15005.9 724.0 (1.22 e-3) 595394.7
16005.6 861.0 (1.29 e-3) 668478.3
17004.4 1011.3 (1.35 e-3) 746898.1
18005.0 1173.8 (1.43 e-3) 818549.5
19003.5 1351.4 (1.49 e-3) 904551.5
20006.1 1556.2 (1.58 e-3) 986185.0
21005.0 1732.7 (1.63 e-3) 1064312.0
22005.3 1909.5 (1.70 e-3) 1123235.3
23005.8 2152.0 (1.77 e-3) 1213077.8
24002.6 2414.8 (1.84 e-3) 1310966.3
25004.9 2666.9 (1.91 e-3) 1399213.0
26005.4 2958.1 (1.98 e-3) 1495500.5
27006.0 3245.6 (2.04 e-3) 1589422.1
28006.0 3560.3 (2.12 e-3) 1677804.0
29005.8 3913.1 (2.19 e-3) 1790073.2
30004.0 4247.4 (2.25 e-3) 1886056.8
31004.0 4618.4 (2.32 e-3) 1987263.3
32004.6 5003.2 (2.39 e-3) 2091638.8
33000.0 5349.2 (2.46 e-3) 2170941.6
34003.2 5781.2 (2.55 e-3) 2265360.5
35004.9 6213.4 (2.60 e-3) 2386098.3
36003.3 6713.6 (2.67 e-3) 2512574.9
37002.1 7305.4 (2.75 e-3) 2656509.1
38005.1 8087.1 (2.81 e-3) 2875924.6
39003.6 8526.0 (2.88 e-3) 2960416.7
39997.8 8835.2 (3.00 e-3) 2943104.6
41005.1 9398.1 (3.02 e-3) 3114015.9
42005.8 9832.8 (3.09 e-3) 3184196.9
43003.1 10339.4 (3.16 e-3) 3267825.5
44005.6 11007.1 (3.23 e-3) 3409882.3
45003.5 11644.2 (3.29 e-3) 3537120.3
46004.8 12352.8 (3.37 e-3) 3667696.0
47005.6 13060.3 (3.44 e-3) 3798807.4
48005.5 13850.3 (3.50 e-3) 3952711.2
49003.3 14825.6 (3.58 e-3) 4145861.3
50005.9 15435.8 (3.65 e-3) 4231304.8
51005.6 16128.1 (3.71 e-3) 4347196.8
52004.8 16998.9 (3.78 e-3) 4497063.5
53005.0 17837.7 (3.85 e-3) 4637987.5
54004.5 18631.3 (3.92 e-3) 4752882.7
55004.8 19613.3 (4.00 e-3) 4903325.0
56004.1 20424.9 (4.06 e-3) 5025812.0
57005.9 21383.3 (4.13 e-3) 5177554.5
58004.2 22470.3 (4.19 e-3) 5357725.3
59003.3 23287.9 (4.27 e-3) 5458954.5
60006.3 24412.8 (4.34 e-3) 5627662.5
61005.9 25196.2 (4.40 e-3) 5726409.1
62003.8 25857.4 (4.48 e-3) 5766592.3
63004.8 26956.6 (4.54 e-3) 5932350.4
64005.4 27839.4 (4.61 e-3) 6038915.4
65005.5 29078.7 (4.70 e-3) 6192227.4
66005.4 29854.0 (4.76 e-3) 6266582.7
67003.5 30966.1 (4.82 e-3) 6424502.1
68006.4 32222.3 (4.90 e-3) 6570615.8
69005.0 33446.1 (4.97 e-3) 6732306.8
70005.6 34635.7 (5.05 e-3) 6861271.8
71005.0 35762.7 (5.10 e-3) 7012294.1
72005.1 36893.8 (5.17 e-3) 7130614.6
73003.9 38375.0 (5.23 e-3) 7334671.3
74005.5 39864.7 (5.32 e-3) 7496182.8

^C
    ***   at top-level: ...errabiniterations,dummy+=ispseudoprime(k));ti
    ***                                             ^--------------------
    *** ispseudoprime: user interrupt after 2h, 29min, 5,695 ms
    ***   Break loop: <Return> to continue; 'break' to go back to GP prompt 
break> break  

[hxlnwdak] human factorization

given a group of people armed with paper and pencils, what size number can they (probably) prime factor in a given amount of time?  what algorithms should they use?

[eoaqgkaq] proportion of RSA numbers

define an RSA number to be a number that factors into exactly two prime numbers, the larger being less than twice the smaller.

isrsa(n)=my(f=factorint(n)); my(s=matsize(f)); if(2!=s[1], return(0), if(f[1,2]!=1 || f[2,2]!=1, return(0), if(f[1,1]*2<f[2,1], return(0)))); 1;

here are the first few RSA numbers:

? for(i=0, 256, if(isrsa(i), print(i," ",factorint(i))))
6 [2, 1; 3, 1]
15 [3, 1; 5, 1]
35 [5, 1; 7, 1]
77 [7, 1; 11, 1]
91 [7, 1; 13, 1]
143 [11, 1; 13, 1]
187 [11, 1; 17, 1]
209 [11, 1; 19, 1]
221 [13, 1; 17, 1]
247 [13, 1; 19, 1]

for a given range of numbers, what proportion are RSA numbers?  this seems like this could be estimated analytically (perhaps with the Dickman function, for related work, see Tao), but we do Monte Carlo in Pari/GP, using parfor for parallel computation.

in order to speed up computation by a factor of 2, we assume all even numbers are not RSA numbers.  however, when testing small numbers, 6 needs to be special cased.  even though printing is done in the sequential block of parfor, the results do not come out in order.

the first few results below can be verified with the above list of RSA numbers.  among the 2-bit numbers with most significant bit set, namely 2 and 3, none are RSA numbers.  among the 5-bit numbers with MSB set, 16-31, none are RSA numbers.  among the 32 6-bit numbers with MSB set, 32-63, 1 (35) is an RSA number: 1e6/32 = 31250.  among the 8 4-bit numbers with MSB set, 8-15, 1 (15) is an RSA number: 1e6/8 = 125000.  among the 128 8-bit numbers with MSB set, 128-255, 5 (143, 187, 209, 221, 247) are RSA numbers: 1e6/128*5 = 39062.5.  among the 1-bit number with MSB set, 1, none are RSA numbers.  among the 64 7-bit numbers with MSB set, 64-127, 2 (35, 77) are RSA numbers: 1e6/64*2 = 31250.  among the 4 3-bit numbers with MSB set, 4-7, 1 (6) is an RSA number: 1e6/4 = 250000.

first batch (actually done second), up to 55 bits:

? gettime; export(isrsa); my(ncount=1000000); parfor(bitsize=1, 55, my(s=0); for(i=1, ncount, my(n=random(2^(bitsize-1))+2^(bitsize-1)); if((n!=6)&&(n%2==0), next); if(isrsa(n), s+=1)); s, x, print("dataline "bitsize" "x" "ncount" "gettime))
dataline 2 0 1000000 18573
dataline 5 0 1000000 1904
dataline 6 31267 1000000 721
dataline 4 124853 1000000 161
dataline 8 39461 1000000 429
dataline 1 0 1000000 657
dataline 7 31457 1000000 619
dataline 3 249848 1000000 3514
dataline 10 19328 1000000 15600
dataline 9 19521 1000000 358
dataline 13 15826 1000000 771
dataline 11 18540 1000000 435
dataline 12 17816 1000000 656
dataline 15 11376 1000000 1572
dataline 14 12527 1000000 70
dataline 16 10891 1000000 2589
dataline 17 9477 1000000 15102
dataline 20 7257 1000000 2855
dataline 18 8500 1000000 1187
dataline 19 7705 1000000 1742
dataline 21 6747 1000000 2350
dataline 22 5958 1000000 1498
dataline 23 5487 1000000 2263
dataline 24 5089 1000000 2358
dataline 25 4625 1000000 15038
dataline 26 4297 1000000 5640
dataline 27 3963 1000000 2492
dataline 28 3790 1000000 3167
dataline 29 3613 1000000 3506
dataline 30 3317 1000000 6984
dataline 31 3043 1000000 6711
dataline 32 2769 1000000 8230
dataline 33 2616 1000000 23971
dataline 34 2583 1000000 18296
dataline 35 2467 1000000 10707
dataline 36 2216 1000000 8008
dataline 37 2173 1000000 17621
dataline 38 2078 1000000 16277
dataline 39 1908 1000000 25297
dataline 40 1863 1000000 22039
dataline 41 1790 1000000 39006
dataline 42 1694 1000000 42367
dataline 43 1593 1000000 38054
dataline 44 1524 1000000 35728
dataline 45 1499 1000000 49830
dataline 46 1410 1000000 48127
dataline 47 1351 1000000 62108
dataline 48 1282 1000000 65871
dataline 49 1188 1000000 76204
dataline 50 1231 1000000 71772
dataline 51 1122 1000000 66634
dataline 52 1074 1000000 44500
dataline 53 1029 1000000 56539
dataline 54 1007 1000000 37354
dataline 55 954 1000000 22767
cpu time = 17min, 8,807 ms, real time = 5min, 50,115 ms.

for example, the last line above should be read: we randomly sampled 1000000 55-bit numbers with MSB set.  954 were RSA numbers = 0.000954 = 0.0954% .  22767 milliseconds of CPU time had been expended since the previous line was printed (though gettime is mostly meaningless in this parallel computation).

second batch:

? gettime; export(isrsa); my(ncount=1000000); parfor(bitsize=56, +oo, my(s=0); for(i=1, ncount, my(n=random(2^(bitsize-1))+2^(bitsize-1)); if(n%2==0, next); if(isrsa(n), s+=1)); s, x, print("dataline "bitsize" "x" "ncount" "gettime))
dataline 56 928 1000000 967649
dataline 57 913 1000000 139615
dataline 58 887 1000000 154693
dataline 60 841 1000000 74465
dataline 59 837 1000000 107296
dataline 61 850 1000000 11812
dataline 62 763 1000000 70641
dataline 63 726 1000000 78389
dataline 64 722 1000000 1068514
dataline 65 690 1000000 446623
dataline 66 638 1000000 266677
dataline 67 694 1000000 211371
dataline 68 641 1000000 253129
dataline 69 619 1000000 164697
dataline 70 557 1000000 221582
dataline 71 577 1000000 253060
dataline 72 583 1000000 1208599
dataline 73 511 1000000 584578
dataline 74 502 1000000 448724
dataline 75 498 1000000 380748
dataline 76 499 1000000 391445
dataline 77 470 1000000 382709
dataline 78 471 1000000 403028
dataline 79 483 1000000 458870
dataline 80 495 1000000 1413907
dataline 81 431 1000000 791296
dataline 82 436 1000000 694658
dataline 83 411 1000000 649589
dataline 84 412 1000000 550870
dataline 85 399 1000000 621967
dataline 86 407 1000000 650454
dataline 87 399 1000000 717952
dataline 88 392 1000000 1735925
dataline 89 342 1000000 1041960
dataline 90 352 1000000 996877
dataline 91 346 1000000 1002713
dataline 92 341 1000000 860047
dataline 93 304 1000000 955470
dataline 94 333 1000000 1012760
dataline 95 329 1000000 1054538
dataline 96 299 1000000 2080883
dataline 97 299 1000000 1574054
dataline 98 283 1000000 1497025
dataline 99 300 1000000 1541806
dataline 100 313 1000000 1266607
dataline 101 287 1000000 1577068
dataline 102 289 1000000 1557525
dataline 103 265 1000000 1650203
dataline 104 299 1000000 2721996
dataline 105 268 1000000 2171018
dataline 106 261 1000000 2138428
dataline 107 248 1000000 2464619
dataline 108 273 1000000 2069371
dataline 109 233 1000000 2549832
dataline 110 228 1000000 2507602
dataline 111 242 1000000 2683309
dataline 112 225 1000000 3787277
dataline 113 223 1000000 3385594
dataline 114 181 1000000 3457664
dataline 115 212 1000000 3773621
dataline 116 235 1000000 3241294
dataline 117 212 1000000 4039748
dataline 118 196 1000000 4076002
dataline 119 197 1000000 4216571
dataline 120 191 1000000 6046672
dataline 121 205 1000000 5242164
dataline 122 193 1000000 5826027
dataline 123 184 1000000 5985150
dataline 124 193 1000000 5220699
dataline 125 183 1000000 6394398
dataline 126 166 1000000 6804860
dataline 127 182 1000000 7052941
dataline 128 199 1000000 9081431
dataline 129 164 1000000 9493346
dataline 130 161 1000000 9527733
dataline 131 151 1000000 10311288
dataline 132 177 1000000 9186555
dataline 133 169 1000000 11124547
dataline 134 178 1000000 12562868
dataline 135 154 1000000 12741479
dataline 136 167 1000000 15479663
dataline 137 157 1000000 15436601
dataline 138 155 1000000 16690681
dataline 139 148 1000000 17277977
dataline 140 137 1000000 19504895
dataline 141 162 1000000 18781695
dataline 142 138 1000000 22060326
dataline 143 152 1000000 22222123
dataline 144 134 1000000 26356486
dataline 145 150 1000000 24985983
dataline 146 159 1000000 27665757
dataline 147 155 1000000 29276993
dataline 148 124 1000000 31849872
dataline 149 127 1000000 30683069
dataline 150 126 1000000 35491505
*** parfor: user interrupt after 163h, 51min, 56,082 ms cpu time, 21h, 51min, 54,044 ms real time

[iufsumdd] consecutive differences of primes

finite differences are a common way to analyze sequences of numbers.  we compute first, second,... twentieth differences of prime numbers in various ranges.

finite differences work well for analysis when the sequence grows polynomially or exponentially, but primes do neither.  the magnitudes of high-order differences get large.  the first row is OEIS A007442.  here is a plot of A007442 divided by (-2)^n (following the suggestion in the OEIS entry).  it is surprising that something so jagged and random as the prime numbers can result in such a smooth plot.

vaguely similar but totally different: Riemann's prime counting function is another way of arriving at prime numbers through an infinite sum.

the entries in the tables below satisfy A + B = C.

A	B
C

primes 2 through 541:

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	1	1	-1	3	-9	23	-53	115	-237	457	-801	1213	-1389	445	3667	-15081	41335	-95059	195769	-370803
3	2	0	2	-6	14	-30	62	-122	220	-344	412	-176	-944	4112	-11414	26254	-53724	100710	-175034	281660
5	2	2	-4	8	-16	32	-60	98	-124	68	236	-1120	3168	-7302	14840	-27470	46986	-74324	106626	-129236
7	4	-2	4	-8	16	-28	38	-26	-56	304	-884	2048	-4134	7538	-12630	19516	-27338	32302	-22610	-33286
11	2	2	-4	8	-12	10	12	-82	248	-580	1164	-2086	3404	-5092	6886	-7822	4964	9692	-55896	177960
13	4	-2	4	-4	-2	22	-70	166	-332	584	-922	1318	-1688	1794	-936	-2858	14656	-46204	122064	-288534
17	2	2	0	-6	20	-48	96	-166	252	-338	396	-370	106	858	-3794	11798	-31548	75860	-166470	334970
19	4	2	-6	14	-28	48	-70	86	-86	58	26	-264	964	-2936	8004	-19750	44312	-90610	168500	-281884
23	6	-4	8	-14	20	-22	16	0	-28	84	-238	700	-1972	5068	-11746	24562	-46298	77890	-113384	129766
29	2	4	-6	6	-2	-6	16	-28	56	-154	462	-1272	3096	-6678	12816	-21736	31592	-35494	16382	60802
31	6	-2	0	4	-8	10	-12	28	-98	308	-810	1824	-3582	6138	-8920	9856	-3902	-19112	77184	-198960
37	4	-2	4	-4	2	-2	16	-70	210	-502	1014	-1758	2556	-2782	936	5954	-23014	58072	-121776	227164
41	2	2	0	-2	0	14	-54	140	-292	512	-744	798	-226	-1846	6890	-17060	35058	-63704	105388	-162066
43	4	2	-2	-2	14	-40	86	-152	220	-232	54	572	-2072	5044	-10170	17998	-28646	41684	-56678	74592
47	6	0	-4	12	-26	46	-66	68	-12	-178	626	-1500	2972	-5126	7828	-10648	13038	-14994	17914	-23138
53	6	-4	8	-14	20	-20	2	56	-190	448	-874	1472	-2154	2702	-2820	2390	-1956	2920	-5224	-4070
59	2	4	-6	6	0	-18	58	-134	258	-426	598	-682	548	-118	-430	434	964	-2304	-9294	83794
61	6	-2	0	6	-18	40	-76	124	-168	172	-84	-134	430	-548	4	1398	-1340	-11598	74500	-288154
67	4	-2	6	-12	22	-36	48	-44	4	88	-218	296	-118	-544	1402	58	-12938	62902	-213654	611686
71	2	4	-6	10	-14	12	4	-40	92	-130	78	178	-662	858	1460	-12880	49964	-150752	398032	-966032
73	6	-2	4	-4	-2	16	-36	52	-38	-52	256	-484	196	2318	-11420	37084	-100788	247280	-568000	1247046
79	4	2	0	-6	14	-20	16	14	-90	204	-228	-288	2514	-9102	25664	-63704	146492	-320720	679046	-1402164
83	6	2	-6	8	-6	-4	30	-76	114	-24	-516	2226	-6588	16562	-38040	82788	-174228	358326	-723118	1431996
89	8	-4	2	2	-10	26	-46	38	90	-540	1710	-4362	9974	-21478	44748	-91440	184098	-364792	708878	-1345738
97	4	-2	4	-8	16	-20	-8	128	-450	1170	-2652	5612	-11504	23270	-46692	92658	-180694	344086	-636860	1143154
101	2	2	-4	8	-4	-28	120	-322	720	-1482	2960	-5892	11766	-23422	45966	-88036	163392	-292774	506294	-848198
103	4	-2	4	4	-32	92	-202	398	-762	1478	-2932	5874	-11656	22544	-42070	75356	-129382	213520	-341904	542648
107	2	2	8	-28	60	-110	196	-364	716	-1454	2942	-5782	10888	-19526	33286	-54026	84138	-128384	200744	-344574
109	4	10	-20	32	-50	86	-168	352	-738	1488	-2840	5106	-8638	13760	-20740	30112	-44246	72360	-143830	339910
113	14	-10	12	-18	36	-82	184	-386	750	-1352	2266	-3532	5122	-6980	9372	-14134	28114	-71470	196080	-524184
127	4	2	-6	18	-46	102	-202	364	-602	914	-1266	1590	-1858	2392	-4762	13980	-43356	124610	-328104	802104
131	6	-4	12	-28	56	-100	162	-238	312	-352	324	-268	534	-2370	9218	-29376	81254	-203494	474000	-1044696
137	2	8	-16	28	-44	62	-76	74	-40	-28	56	266	-1836	6848	-20158	51878	-122240	270506	-570696	1158050
139	10	-8	12	-16	18	-14	-2	34	-68	28	322	-1570	5012	-13310	31720	-70362	148266	-300190	587354	-1113058
149	2	4	-4	2	4	-16	32	-34	-40	350	-1248	3442	-8298	18410	-38642	77904	-151924	287164	-525704	930130
151	6	0	-2	6	-12	16	-2	-74	310	-898	2194	-4856	10112	-20232	39262	-74020	135240	-238540	404426	-655310
157	6	-2	4	-6	4	14	-76	236	-588	1296	-2662	5256	-10120	19030	-34758	61220	-103300	165886	-250884	348802
163	4	2	-2	-2	18	-62	160	-352	708	-1366	2594	-4864	8910	-15728	26462	-42080	62586	-84998	97918	-67186
167	6	0	-4	16	-44	98	-192	356	-658	1228	-2270	4046	-6818	10734	-15618	20506	-22412	12920	30732	-167200
173	6	-4	12	-28	54	-94	164	-302	570	-1042	1776	-2772	3916	-4884	4888	-1906	-9492	43652	-136468	371942
179	2	8	-16	26	-40	70	-138	268	-472	734	-996	1144	-968	4	2982	-11398	34160	-92816	235474	-561868
181	10	-8	10	-14	30	-68	130	-204	262	-262	148	176	-964	2986	-8416	22762	-58656	142658	-326394	702848
191	2	2	-4	16	-38	62	-74	58	0	-114	324	-788	2022	-5430	14346	-35894	84002	-183736	376454	-723622
193	4	-2	12	-22	24	-12	-16	58	-114	210	-464	1234	-3408	8916	-21548	48108	-99734	192718	-347168	580184
197	2	10	-10	2	12	-28	42	-56	96	-254	770	-2174	5508	-12632	26560	-51626	92984	-154450	233016	-307648
199	12	0	-8	14	-16	14	-14	40	-158	516	-1404	3334	-7124	13928	-25066	41358	-61466	78566	-74632	11810
211	12	-8	6	-2	-2	0	26	-118	358	-888	1930	-3790	6804	-11138	16292	-20108	17100	3934	-62822	191218
223	4	-2	4	-4	-2	26	-92	240	-530	1042	-1860	3014	-4334	5154	-3816	-3008	21034	-58888	128396	-245706
227	2	2	0	-6	24	-66	148	-290	512	-818	1154	-1320	820	1338	-6824	18026	-37854	69508	-117310	191908
229	4	2	-6	18	-42	82	-142	222	-306	336	-166	-500	2158	-5486	11202	-19828	31654	-47802	74598	-136346
233	6	-4	12	-24	40	-60	80	-84	30	170	-666	1658	-3328	5716	-8626	11826	-16148	26796	-61748	171834
239	2	8	-12	16	-20	20	-4	-54	200	-496	992	-1670	2388	-2910	3200	-4322	10648	-34952	110086	-315814
241	10	-4	4	-4	0	16	-58	146	-296	496	-678	718	-522	290	-1122	6326	-24304	75134	-205728	524208
251	6	0	0	-4	16	-42	88	-150	200	-182	40	196	-232	-832	5204	-17978	50830	-130594	318480	-751336
257	6	0	-4	12	-26	46	-62	50	18	-142	236	-36	-1064	4372	-12774	32852	-79764	187886	-432856	975194
263	6	-4	8	-14	20	-16	-12	68	-124	94	200	-1100	3308	-8402	20078	-46912	108122	-244970	542338	-1168046
269	2	4	-6	6	4	-28	56	-56	-30	294	-900	2208	-5094	11676	-26834	61210	-136848	297368	-625708	1274296
271	6	-2	0	10	-24	28	0	-86	264	-606	1308	-2886	6582	-15158	34376	-75638	160520	-328340	648588	-1241306
277	4	-2	10	-14	4	28	-86	178	-342	702	-1578	3696	-8576	19218	-41262	84882	-167820	320248	-592718	1069624
281	2	8	-4	-10	32	-58	92	-164	360	-876	2118	-4880	10642	-22044	43620	-82938	152428	-272470	476906	-823264
283	10	4	-14	22	-26	34	-72	196	-516	1242	-2762	5762	-11402	21576	-39318	69490	-120042	204436	-346358	588620
293	14	-10	8	-4	8	-38	124	-320	726	-1520	3000	-5640	10174	-17742	30172	-50552	84394	-141922	242262	-420292
307	4	-2	4	4	-30	86	-196	406	-794	1480	-2640	4534	-7568	12430	-20380	33842	-57528	100340	-178030	315634
311	2	2	8	-26	56	-110	210	-388	686	-1160	1894	-3034	4862	-7950	13462	-23686	42812	-77690	137604	-230850
313	4	10	-18	30	-54	100	-178	298	-474	734	-1140	1828	-3088	5512	-10224	19126	-34878	59914	-93246	123654
317	14	-8	12	-24	46	-78	120	-176	260	-406	688	-1260	2424	-4712	8902	-15752	25036	-33332	30408	3790
331	6	4	-12	22	-32	42	-56	84	-146	282	-572	1164	-2288	4190	-6850	9284	-8296	-2924	34198	-88996
337	10	-8	10	-10	10	-14	28	-62	136	-290	592	-1124	1902	-2660	2434	988	-11220	31274	-54798	39974
347	2	2	0	0	-4	14	-34	74	-154	302	-532	778	-758	-226	3422	-10232	20054	-23524	-14824	196310
349	4	2	0	-4	10	-20	40	-80	148	-230	246	20	-984	3196	-6810	9822	-3470	-38348	181486	-576196
353	6	2	-4	6	-10	20	-40	68	-82	16	266	-964	2212	-3614	3012	6352	-41818	143138	-394710	965540
359	8	-2	2	-4	10	-20	28	-14	-66	282	-698	1248	-1402	-602	9364	-35466	101320	-251572	570830	-1212538
367	6	0	-2	6	-10	8	14	-80	216	-416	550	-154	-2004	8762	-26102	65854	-150252	319258	-641708	1231164
373	6	-2	4	-4	-2	22	-66	136	-200	134	396	-2158	6758	-17340	39752	-84398	169006	-322450	589456	-1035236
379	4	2	0	-6	20	-44	70	-64	-66	530	-1762	4600	-10582	22412	-44646	84608	-153444	267006	-445780	711934
383	6	2	-6	14	-24	26	6	-130	464	-1232	2838	-5982	11830	-22234	39962	-68836	113562	-178774	266154	-366930
389	8	-4	8	-10	2	32	-124	334	-768	1606	-3144	5848	-10404	17728	-28874	44726	-65212	87380	-100776	76872
397	4	4	-2	-8	34	-92	210	-434	838	-1538	2704	-4556	7324	-11146	15852	-20486	22168	-13396	-23904	130158
401	8	2	-10	26	-58	118	-224	404	-700	1166	-1852	2768	-3822	4706	-4634	1682	8772	-37300	106254	-258406
409	10	-8	16	-32	60	-106	180	-296	466	-686	916	-1054	884	72	-2952	10454	-28528	68954	-152152	307846
419	2	8	-16	28	-46	74	-116	170	-220	230	-138	-170	956	-2880	7502	-18074	40426	-83198	155694	-260486
421	10	-8	12	-18	28	-42	54	-50	10	92	-308	786	-1924	4622	-10572	22352	-42772	72496	-104792	117610
431	2	4	-6	10	-14	12	4	-40	102	-216	478	-1138	2698	-5950	11780	-20420	29724	-32296	12818	48074
433	6	-2	4	-4	-2	16	-36	62	-114	262	-660	1560	-3252	5830	-8640	9304	-2572	-19478	60892	-95608
439	4	2	0	-6	14	-20	26	-52	148	-398	900	-1692	2578	-2810	664	6732	-22050	41414	-34716	-105642
443	6	2	-6	8	-6	6	-26	96	-250	502	-792	886	-232	-2146	7396	-15318	19364	6698	-140358	579272
449	8	-4	2	2	0	-20	70	-154	252	-290	94	654	-2378	5250	-7922	4046	26062	-133660	438914	-1203534
457	4	-2	4	2	-20	50	-84	98	-38	-196	748	-1724	2872	-2672	-3876	30108	-107598	305254	-764620	1763126
461	2	2	6	-18	30	-34	14	60	-234	552	-976	1148	200	-6548	26232	-77490	197656	-459366	998506	-2061762
463	4	8	-12	12	-4	-20	74	-174	318	-424	172	1348	-6348	19684	-51258	120166	-261710	539140	-1063256	2024980
467	12	-4	0	8	-24	54	-100	144	-106	-252	1520	-5000	13336	-31574	68908	-141544	277430	-524116	961724	-1723724
479	8	-4	8	-16	30	-46	44	38	-358	1268	-3480	8336	-18238	37334	-72636	135886	-246686	437608	-762000	1304500
487	4	4	-8	14	-16	-2	82	-320	910	-2212	4856	-9902	19096	-35302	63250	-110800	190922	-324392	542500	-887224
491	8	-4	6	-2	-18	80	-238	590	-1302	2644	-5046	9194	-16206	27948	-47550	80122	-133470	218108	-344724	514856
499	4	2	4	-20	62	-158	352	-712	1342	-2402	4148	-7012	11742	-19602	32572	-53348	84638	-126616	170132	-180872
503	6	6	-16	42	-96	194	-360	630	-1060	1746	-2864	4730	-7860	12970	-20776	31290	-41978	43516	-10740	-115630
509	12	-10	26	-54	98	-166	270	-430	686	-1118	1866	-3130	5110	-7806	10514	-10688	1538	32776	-126370	348944
521	2	16	-28	44	-68	104	-160	256	-432	748	-1264	1980	-2696	2708	-174	-9150	34314	-93594	222574	-489324
523	18	-12	16	-24	36	-56	96	-176	316	-516	716	-716	12	2534	-9324	25164	-59280	128980	-266750	534838
541	6	4	-8	12	-20	40	-80	140	-200	200	0	-704	2546	-6790	15840	-34116	69700	-137770	268088	-520942

primes after 10^10.  the first prime is 10^10 + 19.

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
+19	14	14	-34	74	-156	314	-592	1028	-1570	1862	-864	-3686	15952	-42942	96314	-196872	386384	-757604	1527168	-3205990
+33	28	-20	40	-82	158	-278	436	-542	292	998	-4550	12266	-26990	53372	-100558	189512	-371220	769564	-1678822	3779078
+61	8	20	-42	76	-120	158	-106	-250	1290	-3552	7716	-14724	26382	-47186	88954	-181708	398344	-909258	2100256	-4819008
+69	28	-22	34	-44	38	52	-356	1040	-2262	4164	-7008	11658	-20804	41768	-92754	216636	-510914	1190998	-2718752	6051552
+97	6	12	-10	-6	90	-304	684	-1222	1902	-2844	4650	-9146	20964	-50986	123882	-294278	680084	-1527754	3332800	-7048330
+103	18	2	-16	84	-214	380	-538	680	-942	1806	-4496	11818	-30022	72896	-170396	385806	-847670	1805046	-3715530	7371614
+121	20	-14	68	-130	166	-158	142	-262	864	-2690	7322	-18204	42874	-97500	215410	-461864	957376	-1910484	3656084	-6683606
+141	6	54	-62	36	8	-16	-120	602	-1826	4632	-10882	24670	-54626	117910	-246454	495512	-953108	1745600	-3027522	4929638
+147	60	-8	-26	44	-8	-136	482	-1224	2806	-6250	13788	-29956	63284	-128544	249058	-457596	792492	-1281922	1902116	-2481448
+207	52	-34	18	36	-144	346	-742	1582	-3444	7538	-16168	33328	-65260	120514	-208538	334896	-489430	620194	-579332	15104
+259	18	-16	54	-108	202	-396	840	-1862	4094	-8630	17160	-31932	55254	-88024	126358	-154534	130764	40862	-564228	1850594
+277	2	38	-54	94	-194	444	-1022	2232	-4536	8530	-14772	23322	-32770	38334	-28176	-23770	171626	-523366	1286366	-2841776
+279	40	-16	40	-100	250	-578	1210	-2304	3994	-6242	8550	-9448	5564	10158	-51946	147856	-351740	763000	-1555410	3010750
+319	24	24	-60	150	-328	632	-1094	1690	-2248	2308	-898	-3884	15722	-41788	95910	-203884	411260	-792410	1455340	-2526072
+343	48	-36	90	-178	304	-462	596	-558	60	1410	-4782	11838	-26066	54122	-107974	207376	-381150	662930	-1070732	1546406
+391	12	54	-88	126	-158	134	38	-498	1470	-3372	7056	-14228	28056	-53852	99402	-173774	281780	-407802	475674	-257188
+403	66	-34	38	-32	-24	172	-460	972	-1902	3684	-7172	13828	-25796	45550	-74372	108006	-126022	67872	218486	-1084136
+469	32	4	6	-56	148	-288	512	-930	1782	-3488	6656	-11968	19754	-28822	33634	-18016	-58150	286358	-865650	2217654
+501	36	10	-50	92	-140	224	-418	852	-1706	3168	-5312	7786	-9068	4812	15618	-76166	228208	-579292	1352004	-3002562
+537	46	-40	42	-48	84	-194	434	-854	1462	-2144	2474	-1282	-4256	20430	-60548	152042	-351084	772712	-1650558	3446082
+583	6	2	-6	36	-110	240	-420	608	-682	330	1192	-5538	16174	-40118	91494	-199042	421628	-877846	1795524	-3578186
+589	8	-4	30	-74	130	-180	188	-74	-352	1522	-4346	10636	-23944	51376	-107548	222586	-456218	917678	-1782662	3278242
+597	4	26	-44	56	-50	8	114	-426	1170	-2824	6290	-13308	27432	-56172	115038	-233632	461460	-864984	1495580	-2289692
+601	30	-18	12	6	-42	122	-312	744	-1654	3466	-7018	14124	-28740	58866	-118594	227828	-403524	630596	-794112	551736
+631	12	-6	18	-36	80	-190	432	-910	1812	-3552	7106	-14616	30126	-59728	109234	-175696	227072	-163516	-242376	1413326
+643	6	12	-18	44	-110	242	-478	902	-1740	3554	-7510	15510	-29602	49506	-66462	51376	63556	-405892	1170950	-2575792
+649	18	-6	26	-66	132	-236	424	-838	1814	-3956	8000	-14092	19904	-16956	-15086	114932	-342336	765058	-1404842	2094204
+667	12	20	-40	66	-104	188	-414	976	-2142	4044	-6092	5812	2948	-32042	99846	-227404	422722	-639784	689362	-66392
+679	32	-20	26	-38	84	-226	562	-1166	1902	-2048	-280	8760	-29094	67804	-127558	195318	-217062	49578	622970	-2418968
+711	12	6	-12	46	-142	336	-604	736	-146	-2328	8480	-20334	38710	-59754	67760	-21744	-167484	672548	-1795998	4015576
+723	18	-6	34	-96	194	-268	132	590	-2474	6152	-11854	18376	-21044	8006	46016	-189228	505064	-1123450	2219578	-3973362
+741	12	28	-62	98	-74	-136	722	-1884	3678	-5702	6522	-2668	-13038	54022	-143212	315836	-618386	1096128	-1753784	2459922
+753	40	-34	36	24	-210	586	-1162	1794	-2024	820	3854	-15706	40984	-89190	172624	-302550	477742	-657656	706138	-279646
+793	6	2	60	-186	376	-576	632	-230	-1204	4674	-11852	25278	-48206	83434	-129926	175192	-179914	48482	426492	-1668808
+799	8	62	-126	190	-200	56	402	-1434	3470	-7178	13426	-22928	35228	-46492	45266	-4722	-131432	474974	-1242316	2849476
+807	70	-64	64	-10	-144	458	-1032	2036	-3708	6248	-9502	12300	-11264	-1226	40544	-136154	343542	-767342	1607160	-3250964
+877	6	0	54	-154	314	-574	1004	-1672	2540	-3254	2798	1036	-12490	39318	-95610	207388	-423800	839818	-1643804	3221678
+883	6	54	-100	160	-260	430	-668	868	-714	-456	3834	-11454	26828	-56292	111778	-216412	416018	-803986	1577874	-3170796
+889	60	-46	60	-100	170	-238	200	154	-1170	3378	-7620	15374	-29464	55486	-104634	199606	-387968	773888	-1592922	3379460
+949	14	14	-40	70	-68	-38	354	-1016	2208	-4242	7754	-14090	26022	-49148	94972	-188362	385920	-819034	1786538	-3934754
+963	28	-26	30	2	-106	316	-662	1192	-2034	3512	-6336	11932	-23126	45824	-93390	197558	-433114	967504	-2148216	4634166
+991	2	4	32	-104	210	-346	530	-842	1478	-2824	5596	-11194	22698	-47566	104168	-235556	534390	-1180712	2485950	-4917578
+993	6	36	-72	106	-136	184	-312	636	-1346	2772	-5598	11504	-24868	56602	-131388	298834	-646322	1305238	-2431628	4126858
+999	42	-36	34	-30	48	-128	324	-710	1426	-2826	5906	-13364	31734	-74786	167446	-347488	658916	-1126390	1695230	-2119298
+1041	6	-2	4	18	-80	196	-386	716	-1400	3080	-7458	18370	-43052	92660	-180042	311428	-467474	568840	-424068	-333104
+1047	4	2	22	-62	116	-190	330	-684	1680	-4378	10912	-24682	49608	-87382	131386	-156046	101366	144772	-757172	1952146
+1051	6	24	-40	54	-74	140	-354	996	-2698	6534	-13770	24926	-37774	44004	-24660	-54680	246138	-612400	1194974	-1924118
+1057	30	-16	14	-20	66	-214	642	-1702	3836	-7236	11156	-12848	6230	19344	-79340	191458	-366262	582574	-729144	471858
+1087	14	-2	-6	46	-148	428	-1060	2134	-3400	3920	-1692	-6618	25574	-59996	112118	-174804	216312	-146570	-257286	1514126
+1101	12	-8	40	-102	280	-632	1074	-1266	520	2228	-8310	18956	-34422	52122	-62686	41508	69742	-403856	1256840	-3275964
+1113	4	32	-62	178	-352	442	-192	-746	2748	-6082	10646	-15466	17700	-10564	-21178	111250	-334114	852984	-2019124	4566958
+1117	36	-30	116	-174	90	250	-938	2002	-3334	4564	-4820	2234	7136	-31742	90072	-222864	518870	-1166140	2547834	-5409622
+1153	6	86	-58	-84	340	-688	1064	-1332	1230	-256	-2586	9370	-24606	58330	-132792	296006	-647270	1381694	-2861788	5721970
+1159	92	28	-142	256	-348	376	-268	-102	974	-2842	6784	-15236	33724	-74462	163214	-351264	734424	-1480094	2860182	-5279604
+1251	120	-114	114	-92	28	108	-370	872	-1868	3942	-8452	18488	-40738	88752	-188050	383160	-745670	1380088	-2419422	3992328
+1371	6	0	22	-64	136	-262	502	-996	2074	-4510	10036	-22250	48014	-99298	195110	-362510	634418	-1039334	1572906	-2131102
+1377	6	22	-42	72	-126	240	-494	1078	-2436	5526	-12214	25764	-51284	95812	-167400	271908	-404916	533572	-558196	234078
+1383	28	-20	30	-54	114	-254	584	-1358	3090	-6688	13550	-25520	44528	-71588	104508	-133008	128656	-24624	-324118	1217504
+1411	8	10	-24	60	-140	330	-774	1732	-3598	6862	-11970	19008	-27060	32920	-28500	-4352	104032	-348742	893386	-2038730
+1419	18	-14	36	-80	190	-444	958	-1866	3264	-5108	7038	-8052	5860	4420	-32852	99680	-244710	544644	-1145344	2324338
+1437	4	22	-44	110	-254	514	-908	1398	-1844	1930	-1014	-2192	10280	-28432	66828	-145030	299934	-600700	1178994	-2296180
+1441	26	-22	66	-144	260	-394	490	-446	86	916	-3206	8088	-18152	38396	-78202	154904	-300766	578294	-1117186	2207230
+1467	4	44	-78	116	-134	96	44	-360	1002	-2290	4882	-10064	20244	-39806	76702	-145862	277528	-538892	1090044	-2323548
+1471	48	-34	38	-18	-38	140	-316	642	-1288	2592	-5182	10180	-19562	36896	-69160	131666	-261364	551152	-1233504	2873568
+1519	14	4	20	-56	102	-176	326	-646	1304	-2590	4998	-9382	17334	-32264	62506	-129698	289788	-682352	1640064	-3919932
+1533	18	24	-36	46	-74	150	-320	658	-1286	2408	-4384	7952	-14930	30242	-67192	160090	-392564	957712	-2279868	5249892
+1551	42	-12	10	-28	76	-170	338	-628	1122	-1976	3568	-6978	15312	-36950	92898	-232474	565148	-1322156	2970024	-6409290
+1593	30	-2	-18	48	-94	168	-290	494	-854	1592	-3410	8334	-21638	55948	-139576	332674	-757008	1647868	-3439266	6894982
+1623	28	-20	30	-46	74	-122	204	-360	738	-1818	4924	-13304	34310	-83628	193098	-424334	890860	-1791398	3455716	-6401512
+1651	8	10	-16	28	-48	82	-156	378	-1080	3106	-8380	21006	-49318	109470	-231236	466526	-900538	1664318	-2945796	4994412
+1659	18	-6	12	-20	34	-74	222	-702	2026	-5274	12626	-28312	60152	-121766	235290	-434012	763780	-1281478	2048616	-3118648
+1677	12	6	-8	14	-40	148	-480	1324	-3248	7352	-15686	31840	-61614	113524	-198722	329768	-517698	767138	-1070032	1400332
+1689	18	-2	6	-26	108	-332	844	-1924	4104	-8334	16154	-29774	51910	-85198	131046	-187930	249440	-302894	330300	-316164
+1707	16	4	-20	82	-224	512	-1080	2180	-4230	7820	-13620	22136	-33288	45848	-56884	61510	-53454	27406	14136	-39452
+1723	20	-16	62	-142	288	-568	1100	-2050	3590	-5800	8516	-11152	12560	-11036	4626	8056	-26048	41542	-25316	-124520
+1743	4	46	-80	146	-280	532	-950	1540	-2210	2716	-2636	1408	1524	-6410	12682	-17992	15494	16226	-149836	629026
+1747	50	-34	66	-134	252	-418	590	-670	506	80	-1228	2932	-4886	6272	-5310	-2498	31720	-133610	479190	-1589168
+1797	16	32	-68	118	-166	172	-80	-164	586	-1148	1704	-1954	1386	962	-7808	29222	-101890	345580	-1109978	3323318
+1813	48	-36	50	-48	6	92	-244	422	-562	556	-250	-568	2348	-6846	21414	-72668	243690	-764398	2213340	-5941088
+1861	12	14	2	-42	98	-152	178	-140	-6	306	-818	1780	-4498	14568	-51254	171022	-520708	1448942	-3727748	8978254
+1873	26	16	-40	56	-54	26	38	-146	300	-512	962	-2718	10070	-36686	119768	-349686	928234	-2278806	5250506	-11492454
+1899	42	-24	16	2	-28	64	-108	154	-212	450	-1756	7352	-26616	83082	-229918	578548	-1350572	2971700	-6241948	12641998
+1941	18	-8	18	-26	36	-44	46	-58	238	-1306	5596	-19264	56466	-146836	348630	-772024	1621128	-3270248	6400050	-12230330
+1959	10	10	-8	10	-8	2	-12	180	-1068	4290	-13668	37202	-90370	201794	-423394	849104	-1649120	3129802	-5830280	10659460
+1969	20	2	2	2	-6	-10	168	-888	3222	-9378	23534	-53168	111424	-221600	425710	-800016	1480682	-2700478	4829180	-8389540
+1989	22	4	4	-4	-16	158	-720	2334	-6156	14156	-29634	58256	-110176	204110	-374306	680666	-1219796	2128702	-3560360	5587466
+2011	26	8	0	-20	142	-562	1614	-3822	8000	-15478	28622	-51920	93934	-170196	306360	-539130	908906	-1431658	2027106	-2365470
+2037	34	8	-20	122	-420	1052	-2208	4178	-7478	13144	-23298	42014	-76262	136164	-232770	369776	-522752	595448	-338364	-787254
+2071	42	-12	102	-298	632	-1156	1970	-3300	5666	-10154	18716	-34248	59902	-96606	137006	-152976	72696	257084	-1125618	3022674
+2113	30	90	-196	334	-524	814	-1330	2366	-4488	8562	-15532	25654	-36704	40400	-15970	-80280	329780	-868534	1897056	-3661212
+2143	120	-106	138	-190	290	-516	1036	-2122	4074	-6970	10122	-11050	3696	24430	-96250	249500	-538754	1028522	-1764156	2696100
+2263	14	32	-52	100	-226	520	-1086	1952	-2896	3152	-928	-7354	28126	-71820	153250	-289254	489768	-735634	931944	-818244
+2277	46	-20	48	-126	294	-566	866	-944	256	2224	-8282	20772	-43694	81430	-136004	200514	-245866	196310	113700	-1014678
+2323	26	28	-78	168	-272	300	-78	-688	2480	-6058	12490	-22922	37736	-54574	64510	-45352	-49556	310010	-900978	2119970
+2349	54	-50	90	-104	28	222	-766	1792	-3578	6432	-10432	14814	-16838	9936	19158	-94908	260454	-590968	1218992	-2384652
+2403	4	40	-14	-76	250	-544	1026	-1786	2854	-4000	4382	-2024	-6902	29094	-75750	165546	-330514	628024	-1165660	2153768
+2407	44	26	-90	174	-294	482	-760	1068	-1146	382	2358	-8926	22192	-46656	89796	-164968	297510	-537636	988108	-1864050
+2451	70	-64	84	-120	188	-278	308	-78	-764	2740	-6568	13266	-24464	43140	-75172	132542	-240126	450472	-875942	1756466
+2521	6	20	-36	68	-90	30	230	-842	1976	-3828	6698	-11198	18676	-32032	57370	-107584	210346	-425470	880524	-1838086
+2527	26	-16	32	-22	-60	260	-612	1134	-1852	2870	-4500	7478	-13356	25338	-50214	102762	-215124	455054	-957562	1971020

primes after 10^100.  the first prime is 10^100 + 267.

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
+267	682	-388	144	740	-2164	3274	-2396	-3076	17284	-47918	111288	-242890	516120	-1070538	2148082	-4133650	7605906	-13450390	23220570	-40241238
+949	294	-244	884	-1424	1110	878	-5472	14208	-30634	63370	-131602	273230	-554418	1077544	-1985568	3472256	-5844484	9770180	-17020668	32327986
+1243	50	640	-540	-314	1988	-4594	8736	-16426	32736	-68232	141628	-281188	523126	-908024	1486688	-2372228	3925696	-7250488	15307318	-35152482
+1293	690	100	-854	1674	-2606	4142	-7690	16310	-35496	73396	-139560	241938	-384898	578664	-885540	1553468	-3324792	8056830	-19845164	46884198
+1983	790	-754	820	-932	1536	-3548	8620	-19186	37900	-66164	102378	-142960	193766	-306876	667928	-1771324	4732038	-11788334	27039034	-57404142
+2773	36	66	-112	604	-2012	5072	-10566	18714	-28264	36214	-40582	50806	-113110	361052	-1103396	2960714	-7056296	15250700	-30365108	56210648
+2809	102	-46	492	-1408	3060	-5494	8148	-9550	7950	-4368	10224	-62304	247942	-742344	1857318	-4095582	8194404	-15114408	25845540	-40663904
+2911	56	446	-916	1652	-2434	2654	-1402	-1600	3582	5856	-52080	185638	-494402	1114974	-2238264	4098822	-6920004	10731132	-14818364	16198006
+2967	502	-470	736	-782	220	1252	-3002	1982	9438	-46224	133558	-308764	620572	-1123290	1860558	-2821182	3811128	-4087232	1379642	10504316
+3469	32	266	-46	-562	1472	-1750	-1020	11420	-36786	87334	-175206	311808	-502718	737268	-960624	989946	-276104	-2707590	11883958	-36586288
+3501	298	220	-608	910	-278	-2770	10400	-25366	50548	-87872	136602	-190910	234550	-223356	29322	713842	-2983694	9176368	-24702330	60901628
+3799	518	-388	302	632	-3048	7630	-14966	25182	-37324	48730	-54308	43640	11194	-194034	743164	-2269852	6192674	-15525962	36199298	-79094940
+4317	130	-86	934	-2416	4582	-7336	10216	-12142	11406	-5578	-10668	54834	-182840	549130	-1526688	3922822	-9333288	20673336	-42895642	83827768
+4447	44	848	-1482	2166	-2754	2880	-1926	-736	5828	-16246	44166	-128006	366290	-977558	2396134	-5410466	11340048	-22222306	40932126	-71069438
+4491	892	-634	684	-588	126	954	-2662	5092	-10418	27920	-83840	238284	-611268	1418576	-3014332	5929582	-10882258	18709820	-30137312	45240592
+5383	258	50	96	-462	1080	-1708	2430	-5326	17502	-55920	154444	-372984	807308	-1595756	2915250	-4952676	7827562	-11427492	15103280	-17176386
+5641	308	146	-366	618	-628	722	-2896	12176	-38418	98524	-218540	434324	-788448	1319494	-2037426	2874886	-3599930	3675788	-2073106	-2880270
+5949	454	-220	252	-10	94	-2174	9280	-26242	60106	-120016	215784	-354124	531046	-717932	837460	-725044	75858	1602682	-4953376	10414688
+6403	234	32	242	84	-2080	7106	-16962	33864	-59910	95768	-138340	176922	-186886	119528	112416	-649186	1678540	-3350694	5461312	-6552268
+6637	266	274	326	-1996	5026	-9856	16902	-26046	35858	-42572	38582	-9964	-67358	231944	-536770	1029354	-1672154	2110618	-1090956	-4867902
+6903	540	600	-1670	3030	-4830	7046	-9144	9812	-6714	-3990	28618	-77322	164586	-304826	492584	-642800	438464	1019662	-5958858	19429608
+7443	1140	-1070	1360	-1800	2216	-2098	668	3098	-10704	24628	-48704	87264	-140240	187758	-150216	-204336	1458126	-4939196	13470750	-32738842
+8583	70	290	-440	416	118	-1430	3766	-7606	13924	-24076	38560	-52976	47518	37542	-354552	1253790	-3481070	8531554	-19268092	41003950
+8653	360	-150	-24	534	-1312	2336	-3840	6318	-10152	14484	-14416	-5458	85060	-317010	899238	-2227280	5050484	-10736538	21735858	-42442144
+9013	210	-174	510	-778	1024	-1504	2478	-3834	4332	68	-19874	79602	-231950	582228	-1328042	2823204	-5686054	10999320	-20706286	38425468
+9223	36	336	-268	246	-480	974	-1356	498	4400	-19806	59728	-152348	350278	-745814	1495162	-2862850	5313266	-9706966	17719182	-32721896
+9259	372	68	-22	-234	494	-382	-858	4898	-15406	39922	-92620	197930	-395536	749348	-1367688	2450416	-4393700	8012216	-15002714	28921958
+9631	440	46	-256	260	112	-1240	4040	-10508	24516	-52698	105310	-197606	353812	-618340	1082728	-1943284	3618516	-6990498	13919244	-28366660
+10071	486	-210	4	372	-1128	2800	-6468	14008	-28182	52612	-92296	156206	-264528	464388	-860556	1675232	-3371982	6928746	-14447416	30473064
+10557	276	-206	376	-756	1672	-3668	7540	-14174	24430	-39684	63910	-108322	199860	-396168	814676	-1696750	3556764	-7518670	16025648	-34221392
+10833	70	170	-380	916	-1996	3872	-6634	10256	-15254	24226	-44412	91538	-196308	418508	-882074	1860014	-3961906	8506978	-18195744	38209820
+10903	240	-210	536	-1080	1876	-2762	3622	-4998	8972	-20186	47126	-104770	222200	-463566	977940	-2101892	4545072	-9688766	20014076	-39644892
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+12817	240	-100	132	126	-1100	3600	-8388	15538	-23832	30284	-29968	16650	14832	-63274	113108	-131392	102650	-202222	1366502	-6847130
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+14287	852	-208	-36	128	-304	804	-1748	2914	-2704	-4988	37850	-138510	396888	-983954	2196972	-4510140	8614120	-15418088	25994624	-41505952
+15139	644	-244	92	-176	500	-944	1166	210	-7692	32862	-100660	258378	-587066	1213018	-2313168	4103980	-6803968	10576536	-15511328	21815504
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+16183	248	-236	240	-120	-222	1598	-6106	17688	-42628	89920	-170970	297264	-474198	690662	-909176	1072580	-1162224	1369384	-2472544	6536900
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+16641	756	-572	558	-498	368	298	-2332	6170	-10416	9864	6690	-58190	171416	-377282	699504	-1128420	1564396	-1693058	698154	3414530
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+17967	192	442	-366	-396	1832	-3810	5998	-7740	7836	-4134	-7198	33378	-89972	214740	-501878	1177962	-2757326	6343264	-14194098	30774684
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+18793	710	-686	674	-542	210	446	-1646	3798	-7630	14848	-30414	68174	-162370	388946	-903280	2006574	-4264896	8729752	-17360784	33855222
+19503	24	-12	132	-332	656	-1200	2152	-3832	7218	-15566	37760	-94196	226576	-514334	1103294	-2258322	4464856	-8631032	16494438	-31438050
+19527	12	120	-200	324	-544	952	-1680	3386	-8348	22194	-56436	132380	-287758	588960	-1155028	2206534	-4166176	7863406	-14943612	28683480
+19539	132	-80	124	-220	408	-728	1706	-4962	13846	-34242	75944	-155378	301202	-566068	1051506	-1959642	3697230	-7080206	13739868	-26967200
+19671	52	44	-96	188	-320	978	-3256	8884	-20396	41702	-79434	145824	-264866	485438	-908136	1737588	-3382976	6659662	-13227332	26603542
+19723	96	-52	92	-132	658	-2278	5628	-11512	21306	-37732	66390	-119042	220572	-422698	829452	-1645388	3276686	-6567670	13376210	-27936040
+19819	44	40	-40	526	-1620	3350	-5884	9794	-16426	28658	-52652	101530	-202126	406754	-815936	1631298	-3290984	6808540	-14559830	31961774
+19863	84	0	486	-1094	1730	-2534	3910	-6632	12232	-23994	48878	-100596	204628	-409182	815362	-1659686	3517556	-7751290	17401944	-38816344
+19947	84	486	-608	636	-804	1376	-2722	5600	-11762	24884	-51718	104032	-204554	406180	-844324	1857870	-4233734	9650654	-21414400	45581788
+20031	570	-122	28	-168	572	-1346	2878	-6162	13122	-26834	52314	-100522	201626	-438144	1013546	-2375864	5416920	-11763746	24167388	-46972542
+20601	448	-94	-140	404	-774	1532	-3284	6960	-13712	25480	-48208	101104	-236518	575402	-1362318	3041056	-6346826	12403642	-22805154	39638394
+21049	354	-234	264	-370	758	-1752	3676	-6752	11768	-22728	52896	-135414	338884	-786916	1678738	-3305770	6056816	-10401512	16833240	-25723886
+21403	120	30	-106	388	-994	1924	-3076	5016	-10960	30168	-82518	203470	-448032	891822	-1627032	2751046	-4344696	6431728	-8890646	11234174
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+21747	280	-118	0	102	566	-3216	9260	-19878	35460	-55008	75618	-92192	97384	-80832	23746	121496	-487276	1413906	-3676210	8835386
+22027	162	-118	102	668	-2650	6044	-10618	15582	-19548	20610	-16574	5192	16552	-57086	145242	-365780	926630	-2262304	5159176	-10861772
+22189	44	-16	770	-1982	3394	-4574	4964	-3966	1062	4036	-11382	21744	-40534	88156	-220538	560850	-1335674	2896872	-5702596	10254192
+22233	28	754	-1212	1412	-1180	390	998	-2904	5098	-7346	10362	-18790	47622	-132382	340312	-774824	1561198	-2805724	4551596	-6851978
+22261	782	-458	200	232	-790	1388	-1906	2194	-2248	3016	-8428	28832	-84760	207930	-434512	786374	-1244526	1745872	-2300382	3470450
+23043	324	-258	432	-558	598	-518	288	-54	768	-5412	20404	-55928	123170	-226582	351862	-458152	501346	-554510	1170068	-4279600
+23367	66	174	-126	40	80	-230	234	714	-4644	14992	-35524	67242	-103412	125280	-106290	43194	-53164	615558	-3109532	10924718
+23433	240	48	-86	120	-150	4	948	-3930	10348	-20532	31718	-36170	21868	18990	-63096	-9970	562394	-2493974	7815186	-20701710
+23673	288	-38	34	-30	-146	952	-2982	6418	-10184	11186	-4452	-14302	40858	-44106	-73066	552424	-1931580	5321212	-12886524	28629412
+23961	250	-4	4	-176	806	-2030	3436	-3766	1002	6734	-18754	26556	-3248	-117172	479358	-1379156	3389632	-7565312	15742888	-30914080
+24211	246	0	-172	630	-1224	1406	-330	-2764	7736	-12020	7802	23308	-120420	362186	-899798	2010476	-4175680	8177576	-15171192	26582514
+24457	246	-172	458	-594	182	1076	-3094	4972	-4284	-4218	31110	-97112	241766	-537612	1110678	-2165204	4001896	-6993616	11411322	-16809800
+24703	74	286	-136	-412	1258	-2018	1878	688	-8502	26892	-66002	144654	-295846	573066	-1054526	1836692	-2991720	4417706	-5398478	3440320
+24777	360	150	-548	846	-760	-140	2566	-7814	18390	-39110	78652	-151192	277220	-481460	782166	-1155028	1425986	-980772	-1958158	11995896
+25137	510	-398	298	86	-900	2426	-5248	10576	-20720	39542	-72540	126028	-204240	300706	-372862	270958	445214	-2938930	10037738	-28139248
+25647	112	-100	384	-814	1526	-2822	5328	-10144	18822	-32998	53488	-78212	96466	-72156	-101904	716172	-2493716	7098808	-18101510	42699380
+25759	12	284	-430	712	-1296	2506	-4816	8678	-14176	20490	-24724	18254	24310	-174060	614268	-1777544	4605092	-11002702	24597870	-51948070
+25771	296	-146	282	-584	1210	-2310	3862	-5498	6314	-4234	-6470	42564	-149750	440208	-1163276	2827548	-6397610	13595168	-27350200	52498778
+26067	150	136	-302	626	-1100	1552	-1636	816	2080	-10704	36094	-107186	290458	-723068	1664272	-3570062	7197558	-13755032	25148578	-44482174
+26217	286	-166	324	-474	452	-84	-820	2896	-8624	25390	-71092	183272	-432610	941204	-1905790	3627496	-6557474	11393546	-19333596	32762096
+26503	120	158	-150	-22	368	-904	2076	-5728	16766	-45702	112180	-249338	508594	-964586	1721706	-2929978	4836072	-7940050	13428500	-24258554
+26623	278	8	-172	346	-536	1172	-3652	11038	-28936	66478	-137158	259256	-455992	757120	-1208272	1906094	-3103978	5488450	-10830054	23389858
+26901	286	-164	174	-190	636	-2480	7386	-17898	37542	-70680	122098	-196736	301128	-451152	697822	-1197884	2384472	-5341604	12559804	-29346424
+27187	122	10	-16	446	-1844	4906	-10512	19644	-33138	51418	-74638	104392	-150024	246670	-500062	1186588	-2957132	7218200	-16786620	36917420
+27309	132	-6	430	-1398	3062	-5606	9132	-13494	18280	-23220	29754	-45632	96646	-253392	686526	-1770544	4261068	-9568420	20130800	-39836258
+27441	126	424	-968	1664	-2544	3526	-4362	4786	-4940	6534	-15878	51014	-156746	433134	-1084018	2490524	-5307352	10562380	-19705458	34489780
+27567	550	-544	696	-880	982	-836	424	-154	1594	-9344	35136	-105732	276388	-650884	1406506	-2816828	5255028	-9143078	14784322	-21978942

primes after 10^1000. the first prime is 10^1000 + 453.

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
+453	904	452	-170	-368	1430	-1916	-2082	22720	-91410	273660	-681192	1467686	-2784698	4663290	-6770484	7931440	-5084486	-9360890	54615980	-180131366
+1357	1356	282	-538	1062	-486	-3998	20638	-68690	182250	-407532	786494	-1317012	1878592	-2107194	1160956	2846954	-14445376	45255090	-125515386	331707202
+2713	1638	-256	524	576	-4484	16640	-48052	113560	-225282	378962	-530518	561580	-228602	-946238	4007910	-11598422	30809714	-80260296	206191816	-516815836
+4351	1382	268	1100	-3908	12156	-31412	65508	-111722	153680	-151556	31062	332978	-1174840	3061672	-7590512	19211292	-49450582	125931520	-310624020	735469902
+5733	1650	1368	-2808	8248	-19256	34096	-46214	41958	2124	-120494	364040	-841862	1886832	-4528840	11620780	-30239290	76480938	-184692500	424845882	-934716762
+7383	3018	-1440	5440	-11008	14840	-12118	-4256	44082	-118370	243546	-477822	1044970	-2642008	7091940	-18618510	46241648	-108211562	240153382	-509870880	1043800230
+10401	1578	4000	-5568	3832	2722	-16374	39826	-74288	125176	-234276	567148	-1597038	4449932	-11526570	27623138	-61969914	131941820	-269717498	533929350	-1029430608
+11979	5578	-1568	-1736	6554	-13652	23452	-34462	50888	-109100	332872	-1029890	2852894	-7076638	16096568	-34346776	69971906	-137775678	264211852	-495501258	910440810
+17557	4010	-3304	4818	-7098	9800	-11010	16426	-58212	223772	-697018	1823004	-4223744	9019930	-18250208	35625130	-67803772	126436174	-231289406	414939552	-729975680
+21567	706	1514	-2280	2702	-1210	5416	-41786	165560	-473246	1125986	-2400740	4796186	-9230278	17374922	-32178642	58632402	-104853232	183650146	-315036128	530945972
+22273	2220	-766	422	1492	4206	-36370	123774	-307686	652740	-1274754	2395446	-4434092	8144644	-14803720	26453760	-46220830	78796914	-131385982	215909844	-354256666
+24493	1454	-344	1914	5698	-32164	87404	-183912	345054	-622014	1120692	-2038646	3710552	-6659076	11650040	-19767070	32576084	-52589068	84523862	-138346822	235727338
+25947	1110	1570	7612	-26466	55240	-96508	161142	-276960	498678	-917954	1671906	-2948524	4990964	-8117030	12809014	-20012984	31934794	-53822960	97380516	-186202538
+27057	2680	9182	-18854	28774	-41268	64634	-115818	221718	-419276	753952	-1276618	2042440	-3126066	4691984	-7203970	11921810	-21888166	43557556	-88822022	175902302
+29737	11862	-9672	9920	-12494	23366	-51184	105900	-197558	334676	-522666	765822	-1083626	1565918	-2511986	4717840	-9966356	21669390	-45264466	87080280	-148328358
+41599	2190	248	-2574	10872	-27818	54716	-91658	137118	-187990	243156	-317804	482292	-946068	2205854	-5248516	11703034	-23595076	41815814	-61248078	58011838
+43789	2438	-2326	8298	-16946	26898	-36942	45460	-50872	55166	-74648	164488	-463776	1259786	-3042662	6454518	-11892042	18220738	-19432264	-3236240	98929002
+46227	112	5972	-8648	9952	-10044	8518	-5412	4294	-19482	89840	-299288	796010	-1782876	3411856	-5437524	6328696	-1211526	-22668504	95692762	-286361196
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+52423	3408	-1372	1212	-1618	1580	1988	-16306	55170	-139090	287274	-490144	642114	-396688	-1134496	6008342	-18762860	49144228	-117644176	266361048	-579336822
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primes in the vicinity of the record prime gap between 18361375334787046697 and 18361375334787046697 + 1550 .

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
-790	90	-48	70	-136	256	-452	752	-1172	1668	-1982	1240	2874	-15480	46284	-111276	233462	-441804	768760	-1250258	1939212
-700	42	22	-66	120	-196	300	-420	496	-314	-742	4114	-12606	30804	-64992	122186	-208342	326956	-481498	688954	-1018858
-658	64	-44	54	-76	104	-120	76	182	-1056	3372	-8492	18198	-34188	57194	-86156	118614	-154542	207456	-329904	670894
-594	20	10	-22	28	-16	-44	258	-874	2316	-5120	9706	-15990	23006	-28962	32458	-35928	52914	-122448	340990	-955984
-574	30	-12	6	12	-60	214	-616	1442	-2804	4586	-6284	7016	-5956	3496	-3470	16986	-69534	218542	-614994	1753530
-544	18	-6	18	-48	154	-402	826	-1362	1782	-1698	732	1060	-2460	26	13516	-52548	149008	-396452	1138536	-3694978
-526	12	12	-30	106	-248	424	-536	420	84	-966	1792	-1400	-2434	13542	-39032	96460	-247444	742084	-2556442	9131948
-514	24	-18	76	-142	176	-112	-116	504	-882	826	392	-3834	11108	-25490	57428	-150984	494640	-1814358	6575506	-22234426
-490	6	58	-66	34	64	-228	388	-378	-56	1218	-3442	7274	-14382	31938	-93556	343656	-1319718	4761148	-15658920	46964426
-484	64	-8	-32	98	-164	160	10	-434	1162	-2224	3832	-7108	17556	-61618	250100	-976062	3441430	-10897772	31305506	-82620256
-420	56	-40	66	-66	-4	170	-424	728	-1062	1608	-3276	10448	-44062	188482	-725962	2465368	-7456342	20407734	-51314750	120062970
-364	16	26	0	-70	166	-254	304	-334	546	-1668	7172	-33614	144420	-537480	1739406	-4990974	12951392	-30907016	68748220	-144084670
-348	42	26	-70	96	-88	50	-30	212	-1122	5504	-26442	110806	-393060	1201926	-3251568	7960418	-17955624	37841204	-75336450	142934220
-306	68	-44	26	8	-38	20	182	-910	4382	-20938	84364	-282254	808866	-2049642	4708850	-9995206	19885580	-37495246	67597770	-117367320
-238	24	-18	34	-30	-18	202	-728	3472	-16556	63426	-197890	526612	-1240776	2659208	-5286356	9890374	-17609666	30102524	-49769550	80116052
-214	6	16	4	-48	184	-526	2744	-13084	46870	-134464	328722	-714164	1418432	-2627148	4604018	-7719292	12492858	-19667026	30346502	-46294444
-208	22	20	-44	136	-342	2218	-10340	33786	-87594	194258	-385442	704268	-1208716	1976870	-3115274	4773566	-7174168	10679476	-15947942	24285760
-186	42	-24	92	-206	1876	-8122	23446	-53808	106664	-191184	318826	-504448	768154	-1138404	1658292	-2400602	3505308	-5268466	8337818	-14112852
-144	18	68	-114	1670	-6246	15324	-30362	52856	-84520	127642	-185622	263706	-370250	519888	-742310	1104706	-1763158	3069352	-5775034	11398398
-126	86	-46	1556	-4576	9078	-15038	22494	-31664	43122	-57980	78084	-106544	149638	-222422	362396	-658452	1306194	-2705682	5623364	-11443734
-40	40	1510	-3020	4502	-5960	7456	-9170	11458	-14858	20104	-28460	43094	-72784	139974	-296056	647742	-1399488	2917682	-5820370	11083894
+0	1550	-1510	1482	-1458	1496	-1714	2288	-3400	5246	-8356	14634	-29690	67190	-156082	351686	-751746	1518194	-2902688	5263524	-9030320
+1550	40	-28	24	38	-218	574	-1112	1846	-3110	6278	-15056	37500	-88892	195604	-400060	766448	-1384494	2360836	-3766796	5462160
+1590	12	-4	62	-180	356	-538	734	-1264	3168	-8778	22444	-51392	106712	-204456	366388	-618046	976342	-1405960	1695364	-1141506
+1602	8	58	-118	176	-182	196	-530	1904	-5610	13666	-28948	55320	-97744	161932	-251658	358296	-429618	289404	553858	-3346488
+1610	66	-60	58	-6	14	-334	1374	-3706	8056	-15282	26372	-42424	64188	-89726	106638	-71322	-140214	843262	-2792630	7743006
+1676	6	-2	52	8	-320	1040	-2332	4350	-7226	11090	-16052	21764	-25538	16912	35316	-211536	703048	-1949368	4950376	-11974124
+1682	4	50	60	-312	720	-1292	2018	-2876	3864	-4962	5712	-3774	-8626	52228	-176220	491512	-1246320	3001008	-7023748	16146790
+1686	54	110	-252	408	-572	726	-858	988	-1098	750	1938	-12400	43602	-123992	315292	-754808	1754688	-4022740	9123042	-20377700
+1740	164	-142	156	-164	154	-132	130	-110	-348	2688	-10462	31202	-80390	191300	-439516	999880	-2268052	5100302	-11254658	24154002
+1904	22	14	-8	-10	22	-2	20	-458	2340	-7774	20740	-49188	110910	-248216	560364	-1268172	2832250	-6154356	12899344	-25981188
+1926	36	6	-18	12	20	18	-438	1882	-5434	12966	-28448	61722	-137306	312148	-707808	1564078	-3322106	6744988	-13081844	24277782
+1962	42	-12	-6	32	38	-420	1444	-3552	7532	-15482	33274	-75584	174842	-395660	856270	-1758028	3422882	-6336856	11195938	-18952518
+2004	30	-18	26	70	-382	1024	-2108	3980	-7950	17792	-42310	99258	-220818	460610	-901758	1664854	-2913974	4859082	-7756580	11921778
+2034	12	8	96	-312	642	-1084	1872	-3970	9842	-24518	56948	-121560	239792	-441148	763096	-1249120	1945108	-2897498	4165198	-5876466
+2046	20	104	-216	330	-442	788	-2098	5872	-14676	32430	-64612	118232	-201356	321948	-486024	695988	-952390	1267700	-1711268	2523192
+2066	124	-112	114	-112	346	-1310	3774	-8804	17754	-32182	53620	-83124	120592	-164076	209964	-256402	315310	-443568	811924	-1840500
+2190	12	2	2	234	-964	2464	-5030	8950	-14428	21438	-29504	37468	-43484	45888	-46438	58908	-128258	368356	-1028576	2599290
+2202	14	4	236	-730	1500	-2566	3920	-5478	7010	-8066	7964	-6016	2404	-550	12470	-69350	240098	-660220	1570714	-3359106
+2216	18	240	-494	770	-1066	1354	-1558	1532	-1056	-102	1948	-3612	1854	11920	-56880	170748	-420122	910494	-1788392	3217334
+2234	258	-254	276	-296	288	-204	-26	476	-1158	1846	-1664	-1758	13774	-44960	113868	-249374	490372	-877898	1428942	-2067754
+2492	4	22	-20	-8	84	-230	450	-682	688	182	-3422	12016	-31186	68908	-135506	240998	-387526	551044	-638812	391364
+2496	26	2	-28	76	-146	220	-232	6	870	-3240	8594	-19170	37722	-66598	105492	-146528	163518	-87768	-247448	1233212

primes in the vicinity of 39433867730216371575457664399, the start of the smallest known nontrivial prime 21-tuplet, a region of maximal density of 21 consecutive primes, from http://www.pzktupel.de/SMArchiv/smadditions.php .  the 21-tuplet runs from +0 to +84.

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
-1638	158	-146	164	-190	268	-490	1022	-1898	2480	-530	-9106	36128	-96650	215554	-430726	803378	-1444378	2576858	-4672694	8728332
-1480	12	18	-26	78	-222	532	-876	582	1950	-9636	27022	-60522	118904	-215172	372652	-641000	1132480	-2095836	4055638	-8061954
-1468	30	-8	52	-144	310	-344	-294	2532	-7686	17386	-33500	58382	-96268	157480	-268348	491480	-963356	1959802	-4006316	8062166
-1438	22	44	-92	166	-34	-638	2238	-5154	9700	-16114	24882	-37886	61212	-110868	223132	-471876	996446	-2046514	4055850	-7763872
-1416	66	-48	74	132	-672	1600	-2916	4546	-6414	8768	-13004	23326	-49656	112264	-248744	524570	-1050068	2009336	-3708022	6637094
-1350	18	26	206	-540	928	-1316	1630	-1868	2354	-4236	10322	-26330	62608	-136480	275826	-525498	959268	-1698686	2929072	-4864586
-1332	44	232	-334	388	-388	314	-238	486	-1882	6086	-16008	36278	-73872	139346	-249672	433770	-739418	1230386	-1935514	2649244
-1288	276	-102	54	0	-74	76	248	-1396	4204	-9922	20270	-37594	65474	-110326	184098	-305648	490968	-705128	713730	293128
-1012	174	-48	54	-74	2	324	-1148	2808	-5718	10348	-17324	27880	-44852	73772	-121550	185320	-214160	8602	1006858	-4249784
-838	126	6	-20	-72	326	-824	1660	-2910	4630	-6976	10556	-16972	28920	-47778	63770	-28840	-205558	1015460	-3242926	8616262
-712	132	-14	-92	254	-498	836	-1250	1720	-2346	3580	-6416	11948	-18858	15992	34930	-234398	809902	-2227466	5373336	-11836898
-580	118	-106	162	-244	338	-414	470	-626	1234	-2836	5532	-6910	-2866	50922	-199468	575504	-1417564	3145870	-6463562	12497858
-462	12	56	-82	94	-76	56	-156	608	-1602	2696	-1378	-9776	48056	-148546	376036	-842060	1728306	-3317692	6034296	-10492760
-450	68	-26	12	18	-20	-100	452	-994	1094	1318	-11154	38280	-100490	227490	-466024	886246	-1589386	2716604	-4458464	7063188
-382	42	-14	30	-2	-120	352	-542	100	2412	-9836	27126	-62210	127000	-238534	420222	-703140	1127218	-1741860	2604724	-3775566
-340	28	16	28	-122	232	-190	-442	2512	-7424	17290	-35084	64790	-111534	181688	-282918	424078	-614642	862864	-1170842	1522904
-312	44	44	-94	110	42	-632	2070	-4912	9866	-17794	29706	-46744	70154	-101230	141160	-190564	248222	-307978	352062	-338274
-268	88	-50	16	152	-590	1438	-2842	4954	-7928	11912	-17038	23410	-31076	39930	-49404	57658	-59756	44084	13788	-160026
-180	38	-34	168	-438	848	-1404	2112	-2974	3984	-5126	6372	-7666	8854	-9474	8254	-2098	-15672	57872	-146238	312362
-142	4	134	-270	410	-556	708	-862	1010	-1142	1246	-1294	1188	-620	-1220	6156	-17770	42200	-88366	166124	-279376
-138	138	-136	140	-146	152	-154	148	-132	104	-48	-106	568	-1840	4936	-11614	24430	-46166	77758	-113252	129634
+0	2	4	-6	6	-2	-6	16	-28	56	-154	462	-1272	3096	-6678	12816	-21736	31592	-35494	16382	60802
+2	6	-2	0	4	-8	10	-12	28	-98	308	-810	1824	-3582	6138	-8920	9856	-3902	-19112	77184	-198876
+8	4	-2	4	-4	2	-2	16	-70	210	-502	1014	-1758	2556	-2782	936	5954	-23014	58072	-121692	225702
+12	2	2	0	-2	0	14	-54	140	-292	512	-744	798	-226	-1846	6890	-17060	35058	-63620	104010	-150184
+14	4	2	-2	-2	14	-40	86	-152	220	-232	54	572	-2072	5044	-10170	17998	-28562	40390	-46174	15126
+18	6	0	-4	12	-26	46	-66	68	-12	-178	626	-1500	2972	-5126	7828	-10564	11828	-5784	-31048	179242
+24	6	-4	8	-14	20	-20	2	56	-190	448	-874	1472	-2154	2702	-2736	1264	6044	-36832	148194	-485740
+30	2	4	-6	6	0	-18	58	-134	258	-426	598	-682	548	-34	-1472	7308	-30788	111362	-337546	845074
+32	6	-2	0	6	-18	40	-76	124	-168	172	-84	-134	514	-1506	5836	-23480	80574	-226184	507528	-823634
+38	4	-2	6	-12	22	-36	48	-44	4	88	-218	380	-992	4330	-17644	57094	-145610	281344	-316106	-391934
+42	2	4	-6	10	-14	12	4	-40	92	-130	162	-612	3338	-13314	39450	-88516	135734	-34762	-708040	3434166
+44	6	-2	4	-4	-2	16	-36	52	-38	32	-450	2726	-9976	26136	-49066	47218	100972	-742802	2726126	-7916794
+50	4	2	0	-6	14	-20	16	14	-6	-418	2276	-7250	16160	-22930	-1848	148190	-641830	1983324	-5190668	12207672
+54	6	2	-6	8	-6	-4	30	8	-424	1858	-4974	8910	-6770	-24778	146342	-493640	1341494	-3207344	7017004	-14380642
+60	8	-4	2	2	-10	26	38	-416	1434	-3116	3936	2140	-31548	121564	-347298	847854	-1865850	3809660	-7363638	13715732
+68	4	-2	4	-8	16	64	-378	1018	-1682	820	6076	-29408	90016	-225734	500556	-1017996	1943810	-3553978	6352094	-11367866
+72	2	2	-4	8	80	-314	640	-664	-862	6896	-23332	60608	-135718	274822	-517440	925814	-1610168	2798116	-5015772	9529876
+74	4	-2	4	88	-234	326	-24	-1526	6034	-16436	37276	-75110	139104	-242618	408374	-684354	1187948	-2217656	4514104	-9829574
+78	2	2	92	-146	92	302	-1550	4508	-10402	20840	-37834	63994	-103514	165756	-275980	503594	-1029708	2296448	-5315470	12269264
+80	4	94	-54	-54	394	-1248	2958	-5894	10438	-16994	26160	-39520	62242	-110224	227614	-526114	1266740	-3019022	6953794	-15356654
+84	98	40	-108	340	-854	1710	-2936	4544	-6556	9166	-13360	22722	-47982	117390	-298500	740626	-1752282	3934772	-8402860	17138350
+182	138	-68	232	-514	856	-1226	1608	-2012	2610	-4194	9362	-25260	69408	-181110	442126	-1011656	2182490	-4468088	8735490	-16407482
+320	70	164	-282	342	-370	382	-404	598	-1584	5168	-15898	44148	-111702	261016	-569530	1170834	-2285598	4267402	-7671992	13361972
+390	234	-118	60	-28	12	-22	194	-986	3584	-10730	28250	-67554	149314	-308514	601304	-1114764	1981804	-3404590	5689980	-9297296
+624	116	-58	32	-16	-10	172	-792	2598	-7146	17520	-39304	81760	-159200	292790	-513460	867040	-1422786	2285390	-3607316	5591400
+740	58	-26	16	-26	162	-620	1806	-4548	10374	-21784	42456	-77440	133590	-220670	353580	-555746	862604	-1321926	1984084	-2869756
+798	32	-10	-10	136	-458	1186	-2742	5826	-11410	20672	-34984	56150	-87080	132910	-202166	306858	-459322	662158	-885672	1024564
+830	22	-20	126	-322	728	-1556	3084	-5584	9262	-14312	21166	-30930	45830	-69256	104692	-152464	202836	-223514	138892	203276
+852	2	106	-196	406	-828	1528	-2500	3678	-5050	6854	-9764	14900	-23426	35436	-47772	50372	-20678	-84622	342168	-883114
+854	108	-90	210	-422	700	-972	1178	-1372	1804	-2910	5136	-8526	12010	-12336	2600	29694	-105300	257546	-540946	1050604
+962	18	120	-212	278	-272	206	-194	432	-1106	2226	-3390	3484	-326	-9736	32294	-75606	152246	-283400	509658	-916934
+980	138	-92	66	6	-66	12	238	-674	1120	-1164	94	3158	-10062	22558	-43312	76640	-131154	226258	-407276	781628
+1118	46	-26	72	-60	-54	250	-436	446	-44	-1070	3252	-6904	12496	-20754	33328	-54514	95104	-181018	374352	-826876
+1164	20	46	12	-114	196	-186	10	402	-1114	2182	-3652	5592	-8258	12574	-21186	40590	-85914	193334	-452524	1081614
+1184	66	58	-102	82	10	-176	412	-712	1068	-1470	1940	-2666	4316	-8612	19404	-45324	107420	-259190	629090	-1484046
+1250	124	-44	-20	92	-166	236	-300	356	-402	470	-726	1650	-4296	10792	-25920	62096	-151770	369900	-854956	1766592
+1374	80	-64	72	-74	70	-64	56	-46	68	-256	924	-2646	6496	-15128	36176	-89674	218130	-485056	911636	-1216368
+1454	16	8	-2	-4	6	-8	10	22	-188	668	-1722	3850	-8632	21048	-53498	128456	-266926	426580	-304732	-1170666
+1470	24	6	-6	2	-2	2	32	-166	480	-1054	2128	-4782	12416	-32450	74958	-138470	159654	121848	-1475398	5825392
+1494	30	0	-4	0	0	34	-134	314	-574	1074	-2654	7634	-20034	42508	-63512	21184	281502	-1353550	4349994	-11645958
+1524	30	-4	-4	0	34	-100	180	-260	500	-1580	4980	-12400	22474	-21004	-42328	302686	-1072048	2996444	-7295964	16075056