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About irregular primes up to 4002

I’ve been using Kummer’s results relating irregular primes to divisibility properties of numerators of the even-index Bernoulli numbers to produce the odd irregular primes up to 4002.  In the range 2520<= p <= 4002, I find 72 irregular primes, the same as in the ~August 1955 paper of Selfridge, Nicol and Vandiver in P.N.A.S.  That paper was extending work of Lehmer&Lehmer and Vandiver in the range 3<=p <= 2520 in two papers on irregular primes and special methods of Kummer and Vandiver to prove special cases of FLT for the so-called irregular primes, which are more involved conceptually than the regular primes (Kummer).

Here’s my list of irregular primes for 2520<=p<=4002, with an even number ‘2k’ such that p divides the numerator of the Bernoulli number B_{2k}:

 

2543 2374
2557 1464
2579 1730
2591 854
2621 1772
2633 1416
2647 1172
2657 710
2663 1244
2671 404
2689 926
2753 482
2767 2528
2777 1600
2789 1984
2791 2554
2833 1832
2857 98
2861 352
2909 400
2927 242
2939 332
2957 138
2999 776
3011 1496
3023 2020
3049 700
3061 2522
3083 1450
3089 1706
3119 1704
3181 3142
3203 2368
3221 98
3229 1634
3257 922
3313 2222
3323 3292
3329 1378
3391 2232
3407 2076
3433 1300
3469 1174
3491 2544
3511 1416
3517 1836
3529 3490
3533 2314
3539 2082
3559 344
3581 1466
3583 1922
3593 360
3607 1976
3613 2082
3617 16
3631 1104
3637 2526
3671 1580
3677 2238
3697 1884
3779 2362
3797 1256
3821 3296
3833 1840
3851 216
3853 748
3881 1686
3917 1490
3967 106
3989 1936
4001 534

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Written by meditationatae

May 11, 2013 at 7:19 pm

Posted in History

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