OFFSET
0,2
COMMENTS
The sequence of Apéry numbers A005259 defined by A005259(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k)^2 satisfies the pair of supercongruences
1) A005259(n*p^r) == A005259(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r
and
2) A005259(n*p^r - 1) == A005259(n*p^(r-1) - 1) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r.
We conjecture that the present sequence satisfies the same pair of supercongruences. Some examples are given below.
FORMULA
Examples of supercongruences:
a(11) - a(1) = 380777119628689277809 - 7 = 2*(3^3)*7*(11^3)*(19^2)*83*103*587*417773 == 0 (mod 11^3).
a(10) - a(0) = 3026463150129458557 - 1 = (2^2)*3*(11^3)*17*19*191*251*12236761 == 0 (mod 11^3).
MAPLE
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Sep 24 2024
STATUS
approved