[seqfan] "Giuga-Wilson" conjecture and pseudoprimes
Tomasz Ordowski
tomaszordowski at gmail.com
Sun Jun 30 19:23:21 CEST 2019
Dear SeqFans,
I formulated the "Giuga-Wilson" conjecture:
If p is an odd prime, then G(p) - W(p) == 1 (mod p),
where G(p) = (1+Sum_{k=1..p-1}k^{p-1})/p is the Giuga quotient,
and W(p) = (1+(p-1)!)/p is the Wilson quotient A007619(n) for n > 1.
Equivalently:
If p is an odd prime, then Sum_{k=1..p-1}k^{p-1} - (p-1)! == p (mod p^2).
Note: Amiram Eldar found such composite numbers p = 77, 161, 2261.
These numbers can be given the name "Giuga-Wilson" pseudoprimes.
Maybe someone will find more such new pseudoprimes. Really worth.
I am asking for a proof of my conjecture (maybe it is a known theorem).
Best regards,
Thomas Ordowski
____________________
https://oeis.org/A007619
https://oeis.org/A002068
https://oeis.org/A007540
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