[seqfan] Re: (2p-1)!/(p-1)!^2 == p (mod p^4)

[Ami Eldar]

Thomas,
The conjecture is called "The converse of Wolstenholme's theorem". It is
believed to be true, but I think that it was not yet proven. See for
example:
Trevisan, Vilmar, and Kenneth Weber. "Testing the converse of
Wolstenholme's theorem." *Matemática contemporânea. Rio de Janeiro* (2001).
https://www.lume.ufrgs.br/bitstream/handle/10183/448/000317407.pdf
wherethey prove that n cannot be even, and put bounds on it otherwise.

On Fri, Jul 27, 2018 at 6:17 PM, Tomasz Ordowski <tomaszordowski at gmail.com>
wrote:

> P.S. Supplement.
>
> The conjecture:
>
> For n > 3, a(n-1) == 1 (mod n^3) if and only if n is prime,
> where a(n-1) = binomial(2n-1,n) = A001700(n-1).
>
> See https://oeis.org/A001700
>
> The question is how to prove it?
>
> Thomas
>
>
> 2018-07-27 14:11 GMT+02:00 Tomasz Ordowski <tomaszordowski at gmail.com>:
>
> > Dear SeqFan,
> >
> > Conjecture: For n > 3, binomial(2n-1,n) == 1 (mod n^3) if and only if n
> is
> > prime.
> >
> > Equivalently: (2p-1)!/(p-1)!^2 == p (mod p^4) only for all primes p > 3.
> > No proof.
> >
> > Thus a(p-1) == p (mod p^4) for every prime p > 3, where a(n) =
> A002457(n).
> >
> > See https://oeis.org/A002457
> >
> > Question: is a proof of this conjecture known?
> >
> > Does this result from Wilson's theorem?
> >
> > Best regards,
> >
> > Thomas
> >
> >
> >
> >
> >
> > <#m_2849649430454898626_m_1360274943545487338_DAB4FAD8-
> 2DD7-40BB-A1B8-4E2AA1F9FDF2>
> >
>
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