Congruences for sums of binomial coefficients

Details Congruences for sums of binomial coefficients Congruences for sums of binomial coefficients

2005-02-09

arxiv.org/abs/math/0502187v4

J. Number Theory 126(2007), no.2, 287-296

Mathematics

Number Theory

Scientific paper

Let q>1 and m>0 be relatively prime integers. We find an explicit period $\nu_m(q)$ such that for any integers n>0 and r we have $[n+\nu_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q), or a=1 (mod q) and 2|m, where $[n,r]_m(a)=\sum_{k=r(mod m)}\binom{n}{k}a^k$. This is a further extension of a congruence of Glaisher.

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