[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.