[seqfan] Re: "Giuga-Wilson" conjecture and pseudoprimes

[Tomasz Ordowski]

P.S. Note:

The "Giuga-Wilson" pseudoprimes p = 77, 161, 2261, 12839, 14231, ...

The "Giuga-Wilson" primes p such that G(p) - W(p) == 1 (mod p^2),
where G(p) = (1+Sum_{k=1..p-1}k^{p-1})/p and W(p) = (1+(p-1)!)/p.
Equivalent condition: Sum_{k=1..p-1}k^{p-1} - (p-1)! == p (mod p^3).
These are the Lerch primes p = 3, 103, 839, 2237, ...
Cf. https://oeis.org/A197632

Thomas Ordowski & Amiram Eldar
________
Reference: On page 245 of the work by Sondow
https://link.springer.com/chapter/10.1007%2F978-1-4939-1601-6_17
is a new "Lerch's formula" from the proven "Giuga-Wilson" conjecture:
If p is an odd prime, then Sum_{k=1..p-1}k^{p-1} - (p-1)! == p (mod p^2).


niedz., 30 cze 2019 o 19:23 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> 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
>