Bernoulli numbers and arithmetic progressions
Hans Havermann
hahaj at rogers.com
Mon Feb 9 04:27:53 CET 2004
I asked:
> What are the values of p*k-(p-1)/2 where
> Mod[Numerator[BernoulliB[2k]], p] == 0 for p=37?
>
> {574, 1240, 1351, 1906, 2572, 2720, 3238, 3904, 4089, 4570, 5236,
> 5458, 5902, 6568, 6827, 7234, 7900, 8196, 8566, 9232, 9565, 9898, ...}
>
> Every third term in this calculated sequence, it seems, is a number
> that does not figure in the *actual* 37-appearance sequence.
I was still convinced however, ignoring every third term, that 37
appears periodically with a difference of 666 between adjacent terms.
As I calculated and collated the differences between adjacent terms of
a range of irregular primes, hoping to differentiate the clearly
periodic ones from the aberrations, I ran into a rude surprise as I
upped my values of k. Same calculation as above, ignoring every third
term, the differences between adjacent terms for p=37 are:
{666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666,
666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666,
666, 666, 666, 1332, 666, 666, 666, 1332, 111, 1221, 148, 1184, 185,
1147, 222, 1110, 259, 1073, 296, 1036, 333, 999, 370, 962, 407, ...}
I feel a little discombobulated and would appreciate someone confirming
or denying this progression.
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