[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
More information about the SeqFan
mailing list