[seqfan] Is it true?

Tomasz Ordowski tomaszordowski at gmail.com
Mon Jul 11 16:47:06 CEST 2022

 Dear readers,

I found out from an uncertain source that:
(*) If S is a Sierpinski number, then S^p is also a Sierpinski number for
every prime p > 3.
(**) If R is a Riesel number, then R^p is also a Riesel number for every
prime p > 3.
Cf. Wesolowski's comments on A076336 * and A101036 **.
See A076336 - OEIS <http://oeis.org/A076336> and A101036 - OEIS
I am asking for better references.

Therefore, the following conjectures can be put forward:
(1) There are Sierpinski numbers S such that ((2S)^n+1)/(2S+1) is composite
for every odd n > 3.
(2) There are Riesel numbers R such that ((2R)^n-1)/(2R-1) is composite for
every n > 3.
Note that the above formulas can give primes only for prime numbers n.
Which candidates are easy to eliminate numerically?
And which of the rest are provable?

Best regards,

Thomas Ordowski

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