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

[Tomasz Ordowski]

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
>
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