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

[Tomasz Ordowski]

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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