# [seqfan] Primes p = nk-1 dividing Fibonacci( k )

Maximilian Hasler maximilian.hasler at gmail.com
Thu Nov 26 00:33:27 CET 2009

```Dear SeqFans,

The prime p=2269, in which no digit is coprime to its neighbours, is
the least power of 2 to have 6 consecutive '6' in its decimal
representation.

Of course this is completely uninteresting... ;-)

But it is also the norm of the 3rd Eisenstein-Mersenne prime (cf.
Sloane's A066408), and the least prime (of the form p=7k+1) which
divides Fibonacci((p-1)/7).

This last property inspired me the sequence

A168171 Least prime p = 1 (mod n) which divides Fibonacci((p-1)/n)
5,29,139,61,211,541,2269,89,199,281,859,661,911,2269,2221,2081,2789,
2161,3041,421,2521,19009,21529,3001,9901,5981,2161,2269,26449,2221,
31249,19681,17491,2789,3571,25309,30859,3041,6709,3001,9349,2521,13159

(Artiads: p | F[(p-1)/5]), A125253 (p | F[(p-1)/7]), A125252 (p |
F[(p+1)/7]).

However, for the

Least prime p = -1 (mod n) which divides Fibonacci((p+1)/n), or 0 if
there is no such prime < 99999
I get

2, 13, 47, 0, 0, 113, 307, 0, 233, 0, 967, 0, 2417, 797, 0, 0, 1087,
233, 5737, 0, 5417, 5653, 1103, 0, 0, 2417, 4373, 0, 6263, ...

Has anyone a simple and/or universal explanation for the 0's ?

Thanks,
Maximilian

```