we crudely attempt to extend OEIS A051342, the smallest prime factor of P# + 1, where P# is the primorial function.
some factors were found with Pari/GP (using factorint flag 11 to stop after Pollard rho) and GMP-ECM (-one flag). other factors were gathered from https://oeis.org/A051342/b051342.txt , https://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha102.htm , and http://factordb.com . https://oeis.org/A038507/a038507.txt might have been useful, but we did not use it.
we write "crudely attempt to extend" above because the prime factors listed below are not guaranteed to be smallest factors. because both Pollard rho and the Elliptic Curve Method are probabilistic, it is possible that a larger factor was found first by luck, stopping the search, and a smaller factor exists but is not given.
these numbers are interesting because of Euclid's proof of the infinitude of primes. let p be the hypothetical last prime. compute the primorial of p, add one, and factor the number. its smallest prime factor is guaranteed to be larger than p, so, by contradiction, there is no last prime number. the smallest prime factor might be the immediate next prime number after p. this happens at indices 1, 7, 232, and 430, and no known further values (OEIS A081618 and OEIS A058233). another extreme possibility is that the smallest prime factor might be the primorial plus one, making it a "primorial prime".
Euclid's proof also works if you subtract one (future post).
Euclid's proof has a "poof"-like quality to it. it relies on prime factorization, which is theoretically possible but practically very difficult, and certainly Euclid did not have access to computers and modern algorithms.
previously, factoring Sylvester's sequence, another sequence inspired by Euclid's proof.
we ran GMP-ECM with its autoincrement flag (-I), and for completely unfactored entries below, we report the last B1 it tried. however, it's likely that others have gone much further, judging from the large factors found on factordb.com .
nthprime(1)# + 1 = 2# + 1 = 3 is prime
nthprime(2)# + 1 = 3# + 1 = 7 is prime
nthprime(3)# + 1 = 5# + 1 = 31 is prime
nthprime(4)# + 1 = 7# + 1 = 211 is prime
nthprime(5)# + 1 = 11# + 1 = 2311 is prime
nthprime(6)# + 1 = 13# + 1 = 30031 is divisible by 59
nthprime(7)# + 1 = 17# + 1 is divisible by 19
nthprime(8)# + 1 = 19# + 1 is divisible by 347
nthprime(9)# + 1 = 23# + 1 is divisible by 317
nthprime(10)# + 1 = 29# + 1 is divisible by 331
nthprime(11)# + 1 = 31# + 1 is prime
nthprime(12)# + 1 = 37# + 1 is divisible by 181
nthprime(13)# + 1 = 41# + 1 is divisible by 61
nthprime(14)# + 1 = 43# + 1 is divisible by 167
nthprime(15)# + 1 = 47# + 1 is divisible by 953
nthprime(16)# + 1 = 53# + 1 is divisible by 73
nthprime(17)# + 1 = 59# + 1 is divisible by 277
nthprime(18)# + 1 = 61# + 1 is divisible by 223
nthprime(19)# + 1 = 67# + 1 is divisible by 54730729297
nthprime(20)# + 1 = 71# + 1 is divisible by 1063
nthprime(21)# + 1 = 73# + 1 is divisible by 2521
nthprime(22)# + 1 = 79# + 1 is divisible by 22093
nthprime(23)# + 1 = 83# + 1 is divisible by 265739
nthprime(24)# + 1 = 89# + 1 is divisible by 131
nthprime(25)# + 1 = 97# + 1 is divisible by 2336993
nthprime(26)# + 1 = 101# + 1 is divisible by 960703
nthprime(27)# + 1 = 103# + 1 is divisible by 2297
nthprime(28)# + 1 = 107# + 1 is divisible by 149
nthprime(29)# + 1 = 109# + 1 is divisible by 334507
nthprime(30)# + 1 = 113# + 1 is divisible by 5122427
nthprime(31)# + 1 = 127# + 1 is divisible by 1543
nthprime(32)# + 1 = 131# + 1 is divisible by 1951
nthprime(33)# + 1 = 137# + 1 is divisible by 881
nthprime(34)# + 1 = 139# + 1 is divisible by 678279959005528882498681487
nthprime(35)# + 1 = 149# + 1 is divisible by 87549524399
nthprime(36)# + 1 = 151# + 1 is divisible by 23269086799180847
nthprime(37)# + 1 = 157# + 1 is divisible by 1381
nthprime(38)# + 1 = 163# + 1 is divisible by 1361
nthprime(39)# + 1 = 167# + 1 is divisible by 205590139
nthprime(40)# + 1 = 173# + 1 is divisible by 62614127
nthprime(41)# + 1 = 179# + 1 is divisible by 601
nthprime(42)# + 1 = 181# + 1 is divisible by 107453
nthprime(43)# + 1 = 191# + 1 is divisible by 32999
nthprime(44)# + 1 = 193# + 1 is divisible by 21639496447
nthprime(45)# + 1 = 197# + 1 is divisible by 521831
nthprime(46)# + 1 = 199# + 1 is divisible by 467
nthprime(47)# + 1 = 211# + 1 is divisible by 1051
nthprime(48)# + 1 = 223# + 1 is divisible by 13867889468159
nthprime(49)# + 1 = 227# + 1 is divisible by 3187
nthprime(50)# + 1 = 229# + 1 is divisible by 126173
nthprime(51)# + 1 = 233# + 1 is divisible by 5981
nthprime(52)# + 1 = 239# + 1 is divisible by 313
nthprime(53)# + 1 = 241# + 1 is divisible by 3499
nthprime(54)# + 1 = 251# + 1 is divisible by 2963206058339377313
nthprime(55)# + 1 = 257# + 1 is divisible by 34259
nthprime(56)# + 1 = 263# + 1 is divisible by 271
nthprime(57)# + 1 = 269# + 1 is divisible by 17291
nthprime(58)# + 1 = 271# + 1 is divisible by 15559
nthprime(59)# + 1 = 277# + 1 is divisible by 243233
nthprime(60)# + 1 = 281# + 1 is divisible by 62131
nthprime(61)# + 1 = 283# + 1 is divisible by 307
nthprime(62)# + 1 = 293# + 1 is divisible by 397773559
nthprime(63)# + 1 = 307# + 1 is divisible by 45029582471
nthprime(64)# + 1 = 311# + 1 is divisible by 60722339
nthprime(65)# + 1 = 313# + 1 is divisible by 3761
nthprime(66)# + 1 = 317# + 1 is divisible by 131409329
nthprime(67)# + 1 = 331# + 1 is divisible by 8089
nthprime(68)# + 1 = 337# + 1 is divisible by 38453
nthprime(69)# + 1 = 347# + 1 is divisible by 1579
nthprime(70)# + 1 = 349# + 1 is divisible by 44953
nthprime(71)# + 1 = 353# + 1 is divisible by 2143
nthprime(72)# + 1 = 359# + 1 is divisible by 11041477
nthprime(73)# + 1 = 367# + 1 is divisible by 51420888649203007716619883183
nthprime(74)# + 1 = 373# + 1 is divisible by 4973
nthprime(75)# + 1 = 379# + 1 is prime
nthprime(76)# + 1 = 383# + 1 is divisible by 1252724596359114607416742220222030302294387892643573321852443386423988709173
nthprime(77)# + 1 = 389# + 1 is divisible by 216401
nthprime(78)# + 1 = 397# + 1 is divisible by 8629
nthprime(79)# + 1 = 401# + 1 is divisible by 356647
nthprime(80)# + 1 = 409# + 1 is divisible by 13314787
nthprime(81)# + 1 = 419# + 1 is divisible by 1996109
nthprime(82)# + 1 = 421# + 1 is divisible by 2251
nthprime(83)# + 1 = 431# + 1 is divisible by 215983
nthprime(84)# + 1 = 433# + 1 is divisible by 2089
nthprime(85)# + 1 = 439# + 1 is divisible by 1291
nthprime(86)# + 1 = 443# + 1 is divisible by 463
nthprime(87)# + 1 = 449# + 1 is divisible by 10567
nthprime(88)# + 1 = 457# + 1 is divisible by 136386650718776835437339961915915954570962532986071540875811813284285146821
nthprime(89)# + 1 = 461# + 1 is divisible by 44159768129
nthprime(90)# + 1 = 463# + 1 is divisible by 983
nthprime(91)# + 1 = 467# + 1 is divisible by 718713552547
nthprime(92)# + 1 = 479# + 1 is divisible by 154491616237417891
nthprime(93)# + 1 = 487# + 1 is divisible by 11587
nthprime(94)# + 1 = 491# + 1 is divisible by 14419
nthprime(95)# + 1 = 499# + 1 is divisible by 827
nthprime(96)# + 1 = 503# + 1 is divisible by 75883
nthprime(97)# + 1 = 509# + 1 is divisible by 1283
nthprime(98)# + 1 = 521# + 1 is divisible by 3373
nthprime(99)# + 1 = 523# + 1 is divisible by 1069
nthprime(100)# + 1 = 541# + 1 is divisible by 2879
nthprime(101)# + 1 = 547# + 1 is divisible by 15823
nthprime(102)# + 1 = 557# + 1 is divisible by 1657
nthprime(103)# + 1 = 563# + 1 is divisible by 95257
nthprime(104)# + 1 = 569# + 1 is divisible by 10739627
nthprime(105)# + 1 = 571# + 1 is divisible by 1291798623217
nthprime(106)# + 1 = 577# + 1 is divisible by 10304867
nthprime(107)# + 1 = 587# + 1 is divisible by 1033
nthprime(108)# + 1 = 593# + 1 is divisible by 102553727
nthprime(109)# + 1 = 599# + 1 is divisible by 154006033
nthprime(110)# + 1 = 601# + 1 is divisible by 20549
nthprime(111)# + 1 = 607# + 1 is divisible by 3049
nthprime(112)# + 1 = 613# + 1 is divisible by 4871
nthprime(113)# + 1 = 617# + 1 is divisible by 61229914799
nthprime(114)# + 1 = 619# + 1 is divisible by 262429763
nthprime(115)# + 1 = 631# + 1 is divisible by 601751640263
nthprime(116)# + 1 = 641# + 1 is divisible by 17011
nthprime(117)# + 1 = 643# + 1 is divisible by 10181
nthprime(118)# + 1 = 647# + 1 is divisible by 1615231
nthprime(119)# + 1 = 653# + 1 is divisible by 59499433659858893199724793479
nthprime(120)# + 1 = 659# + 1 is divisible by 673
nthprime(121)# + 1 = 661# + 1 is divisible by 531329254115021
nthprime(122)# + 1 = 673# + 1 is divisible by 146266544521347288021221040588983988031510737033058545309139
nthprime(123)# + 1 = 677# + 1 is divisible by 809
nthprime(124)# + 1 = 683# + 1 is divisible by 9391
nthprime(125)# + 1 = 691# + 1 is divisible by 311183
nthprime(126)# + 1 = 701# + 1 is divisible by 2496066055817115603455833191372456039727457
nthprime(127)# + 1 = 709# + 1 is divisible by 3166561
nthprime(128)# + 1 = 719# + 1 is divisible by 160415355065554648327355039857
nthprime(129)# + 1 = 727# + 1 is divisible by 15231063627439042030627880747151148020250169
nthprime(130)# + 1 = 733# + 1 is divisible by 2207
nthprime(131)# + 1 = 739# + 1 is divisible by 1051
nthprime(132)# + 1 = 743# + 1 is divisible by 8039
nthprime(133)# + 1 = 751# + 1 is divisible by 122011
nthprime(134)# + 1 = 757# + 1 is divisible by 118297
nthprime(135)# + 1 = 761# + 1 is divisible by 1105910694925154516147014061
nthprime(136)# + 1 = 769# + 1 is divisible by 997
nthprime(137)# + 1 = 773# + 1 is divisible by 68281
nthprime(138)# + 1 = 787# + 1 is divisible by 5221793
nthprime(139)# + 1 = 797# + 1 is divisible by 802729387
nthprime(140)# + 1 = 809# + 1 is divisible by 1070471
nthprime(141)# + 1 = 811# + 1 is divisible by 1279
nthprime(142)# + 1 = 821# + 1 is divisible by 163332756949
nthprime(143)# + 1 = 823# + 1 is divisible by 3271
nthprime(144)# + 1 = 827# + 1 is divisible by 32983
nthprime(145)# + 1 = 829# + 1 is divisible by 2749
nthprime(146)# + 1 = 839# + 1 is divisible by 1109
nthprime(147)# + 1 = 853# + 1 is divisible by 78643
nthprime(148)# + 1 = 857# + 1 is divisible by 263586391
nthprime(149)# + 1 = 859# + 1 is unfactored, 359 digits, last B1=3450694
nthprime(150)# + 1 = 863# + 1 is divisible by 26699
nthprime(151)# + 1 = 877# + 1 is unfactored, 365 digits, last B1=3445125
nthprime(152)# + 1 = 881# + 1 is divisible by 57033121
nthprime(153)# + 1 = 883# + 1 is divisible by 7773037
nthprime(154)# + 1 = 887# + 1 is divisible by 9001
nthprime(155)# + 1 = 907# + 1 is divisible by 1259
nthprime(156)# + 1 = 911# + 1 is divisible by 222023
nthprime(157)# + 1 = 919# + 1 is divisible by 71899
nthprime(158)# + 1 = 929# + 1 is divisible by 285227
nthprime(159)# + 1 = 937# + 1 is divisible by 1031
nthprime(160)# + 1 = 941# + 1 is divisible by 154354489
nthprime(161)# + 1 = 947# + 1 is divisible by 1975579442865307041379912825628728735003
nthprime(162)# + 1 = 953# + 1 is divisible by 281297273669
nthprime(163)# + 1 = 967# + 1 is divisible by 263448441828540079
nthprime(164)# + 1 = 971# + 1 is divisible by 613968469
nthprime(165)# + 1 = 977# + 1 is divisible by 6301
nthprime(166)# + 1 = 983# + 1 is divisible by 6390031
nthprime(167)# + 1 = 991# + 1 is divisible by 5217923
nthprime(168)# + 1 = 997# + 1 is divisible by 24889
nthprime(169)# + 1 = 1009# + 1 is divisible by 807688338353313872137119097888421949959
nthprime(170)# + 1 = 1013# + 1 is divisible by 563113
nthprime(171)# + 1 = 1019# + 1 is prime
nthprime(172)# + 1 = 1021# + 1 is prime
nthprime(173)# + 1 = 1031# + 1 is divisible by 65305095341
nthprime(174)# + 1 = 1033# + 1 is divisible by 26711
nthprime(175)# + 1 = 1039# + 1 is divisible by 6995563873663
nthprime(176)# + 1 = 1049# + 1 is divisible by 18691
nthprime(177)# + 1 = 1051# + 1 is divisible by 91943
nthprime(178)# + 1 = 1061# + 1 is divisible by 2029
nthprime(179)# + 1 = 1063# + 1 is divisible by 31063
nthprime(180)# + 1 = 1069# + 1 is divisible by 334475159
nthprime(181)# + 1 = 1087# + 1 is divisible by 15822094041475511
nthprime(182)# + 1 = 1091# + 1 is divisible by 2539
nthprime(183)# + 1 = 1093# + 1 is divisible by 85621
nthprime(184)# + 1 = 1097# + 1 is divisible by 1381
nthprime(185)# + 1 = 1103# + 1 is divisible by 3373
nthprime(186)# + 1 = 1109# + 1 is divisible by 728423
nthprime(187)# + 1 = 1117# + 1 is divisible by 3709
nthprime(188)# + 1 = 1123# + 1 is divisible by 134975455840766265227
nthprime(189)# + 1 = 1129# + 1 is divisible by 938071069161701
nthprime(190)# + 1 = 1151# + 1 is divisible by 10987
nthprime(191)# + 1 = 1153# + 1 is divisible by 29077
nthprime(192)# + 1 = 1163# + 1 is divisible by 49069
nthprime(193)# + 1 = 1171# + 1 is divisible by 5228527
nthprime(194)# + 1 = 1181# + 1 is divisible by 1667
nthprime(195)# + 1 = 1187# + 1 is divisible by 15784959895709359
nthprime(196)# + 1 = 1193# + 1 is divisible by 633250175084951
nthprime(197)# + 1 = 1201# + 1 is divisible by 88266884553823
nthprime(198)# + 1 = 1213# + 1 is divisible by 6133
nthprime(199)# + 1 = 1217# + 1 is divisible by 23584344930234589647359
nthprime(200)# + 1 = 1223# + 1 is divisible by 9403
nthprime(201)# + 1 = 1229# + 1 is divisible by 46993
nthprime(202)# + 1 = 1231# + 1 is divisible by 149197
nthprime(203)# + 1 = 1237# + 1 is divisible by 1663
nthprime(204)# + 1 = 1249# + 1 is divisible by 158826487
nthprime(205)# + 1 = 1259# + 1 is divisible by 1867
nthprime(206)# + 1 = 1277# + 1 is divisible by 3041
nthprime(207)# + 1 = 1279# + 1 is divisible by 3643
nthprime(208)# + 1 = 1283# + 1 is divisible by 59766608773786138159718407
nthprime(209)# + 1 = 1289# + 1 is divisible by 1488503394811
nthprime(210)# + 1 = 1291# + 1 is divisible by 1050826689111457
nthprime(211)# + 1 = 1297# + 1 is divisible by 94841
nthprime(212)# + 1 = 1301# + 1 is divisible by 33614065531
nthprime(213)# + 1 = 1303# + 1 is divisible by 66071991095239
nthprime(214)# + 1 = 1307# + 1 is divisible by 11059
nthprime(215)# + 1 = 1319# + 1 is divisible by 4877
nthprime(216)# + 1 = 1321# + 1 is divisible by 5147
nthprime(217)# + 1 = 1327# + 1 is divisible by 6666059674541554171
nthprime(218)# + 1 = 1361# + 1 is divisible by 3827059
nthprime(219)# + 1 = 1367# + 1 is divisible by 7219
nthprime(220)# + 1 = 1373# + 1 is divisible by 56330327
nthprime(221)# + 1 = 1381# + 1 is divisible by 143997071
nthprime(222)# + 1 = 1399# + 1 is unfactored, 582 digits, last B1=1660737
nthprime(223)# + 1 = 1409# + 1 is divisible by 13033
nthprime(224)# + 1 = 1423# + 1 is divisible by 2609
nthprime(225)# + 1 = 1427# + 1 is unfactored, 591 digits, last B1=1973110
nthprime(226)# + 1 = 1429# + 1 is unfactored, 594 digits, last B1=2837975
nthprime(227)# + 1 = 1433# + 1 is divisible by 7159
nthprime(228)# + 1 = 1439# + 1 is divisible by 53411061810953640281429
nthprime(229)# + 1 = 1447# + 1 is divisible by 23032568245429
nthprime(230)# + 1 = 1451# + 1 is divisible by 43019183951
nthprime(231)# + 1 = 1453# + 1 is divisible by 105707502012798824251499
nthprime(232)# + 1 = 1459# + 1 is divisible by 1471
nthprime(233)# + 1 = 1471# + 1 is divisible by 4436989
nthprime(234)# + 1 = 1481# + 1 is divisible by 1619
nthprime(235)# + 1 = 1483# + 1 is divisible by 101827259750880755032647637
nthprime(236)# + 1 = 1487# + 1 is divisible by 4957
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