Congruences for sums of binomial coefficients Original Research Article

Let q>1 and m>0 be relatively prime integers. We find an explicit period νm(q) such that for any integers n>0 and r we haveimage whenever a is an integer with gcd(1−(−a)m,q)=1, or image, or image and 2m, where image. This is a further extension of a congruence of Glaisher.