0,1

For integers j and k, not necessarily positive, define u(n) = k^2*(k - 1)*binomial(j*n,n) - j^2*(j - 1)*binomial(k*n,n). We conjecture that u(p) == u(1) (mod p^5) for all primes p >= 7. This is essentially the case (j, k) = (3, 2). [Follows from Helou and Terjanian (2008), Section 3, Proposition 2.]

Conjecture: for r >= 2, u(p^r) == u(p^(r-1)) ( mod p^(3*r+3) ) for all primes p >= 5. - Peter Bala, Oct 13 2022

R. R. Aidagulov and M. A. Alekseyev, On p-adic approximation of sums of binomial coefficients, Journal of Mathematical Sciences 233:5 (2018), 626-634; arXiv:1602.02632 [math.NT], 2018.

C. Helou and G. Terjanian, On Wolstenholme’s theorem and its converse, J. Number Theory 128 (2008), 475-499.

Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv preprint arXiv:1111.3057 [math.NT], 2011.

a(n) = 2*A005809(n) - 9*A000984(n).

a(p) == a(1) (mod p^4) for all primes p >= 5 by Meštrović, Section 3, equation 15.

Conjecture: the stronger supercongruence a(p) == a(1) (mod p^5) holds for all primes p >= 7.

The conjecture is true: apply Helou and Terjanian, Section 3, Proposition 2. - Peter Bala, Oct 22 2022

The conjecture was proved by Aidagulov and Alekseyev; see the remarks following Corollary 2. - Peter Bala, Oct 29 2022

seq(2*binomial(3*n, n) - 9*binomial(2*n, n), n = 0..20);

Cf. A000984, A005809, A268589, A357508.

Sequence in context: A132195 A076852 A011996 * A190554 A375971 A273244

Adjacent sequences: A357506 A357507 A357508 * A357510 A357511 A357512

sign,easy

Peter Bala, Oct 01 2022

approved