# [seqfan] A strong question

Tomasz Ordowski tomaszordowski at gmail.com
Sun Nov 25 18:57:36 CET 2018

```Dear SeqFans!

Let's define:

Composite numbers m such that b^(m-1) == 1 (mod (b^2-1)m) has a solution b.

25, 49, 65, 85, 91, 121, 125, 133, 145, 169, 185, 205, 217, 221, 247, 259,
...

Note: If such m exists, then the smallest b is in the range 2 <= b <= m-2.

Conjecture:
If m is a composite number such that b^(m-1) == 1 (mod (b^2-1)m) for some
b,
then m is a strong pseudoprime to some base a in the range 2 <= a <= m-2.

Note that not always the smallest a = b.

The question: Is this a proper subset of A181782 ?

Are my pseudoprimes stronger than the strong pseudoprimes?

Cf. https://oeis.org/A181782 (strong pseudoprimes to some base).

Best regards,

Thomas Ordowski

P.S. Let a(n) be the smallest composite k such that n^(k-1) == 1 (mod
(n^2-1)k), for n > 1.

341, 91, 91, 217, 481, 25, 65, 91, 91, 133, 133, 85, 781, 341, 91, 91, 25,
49, 671, 221, 169, ...

The sequence is not in OEIS.

```