OFFSET
0,2
COMMENTS
Sun and Wan's supercongruence stated below apparently generalizes as follows:
Let m be an integer and k a positive integer. Define u(n) = binomial((m+2)*n,(k+1)*n) - binomial(m,k)*binomial((m+2)*n,n). We conjecture that u(n) == u(1) (mod p^5) for all primes p >= 7. [added 22 Oct 2022: the conjecture is true: apply Helou and Terjanian, 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
LINKS
C. Helou and G. Terjanian, On Wolstenholme’s theorem and its converse, J. Number Theory 128 (2008), 475-499.
Z.-W. Sun and D. Wan, On Fleck quotients, arXiv:math/0603462 [math.NT], 2006-2007.
FORMULA
MAPLE
seq(binomial(4*n, 2*n) - 2*binomial(4*n, n), n = 0..20);
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Peter Bala, Oct 01 2022
STATUS
approved