[seqfan] Generally and Especially

Tomasz Ordowski tomaszordowski at gmail.com
Sat Dec 14 14:34:05 CET 2019


Dear SeqFans,

I have an interesting Conjecture:

Generally; for odd natural numbers n,
n^2 | 2^(2^(n-1)-1)-1 if and only if n*ord_{n}(2) | 2^(n-1)-1.

Especially; for odd primes p,
p^2 | 2^(2^(p-1)-1)-1 if and only if ord_{p}(2) | 2^(p-1)-1.
Cf. A188465 : https://oeis.org/A18846

I am asking for a proof or a counterexample.

Thomas
______________
Denote ord_{n}(2) = A002326((n-1)/2).

Composite numbers m such that m*ord_{m} | 2^(m-1)-1 are 4681, 15841, 42799,
52633, 220729, 647089, 1082401, 1145257, 1969417, 3567481, 4835209,
5049001, 5681809, 6140161, 6334351, 8725753, 10712857, 11777599, 12327121,
13500313, 14709241, 22564081, 22849481, 22953673, 23828017, 27271151,
30576151, 41662297, 45485881, ... [data from Amiram Eldar].
This is a proper subset of the pseudoprimes https://oeis.org/A001567



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