Some divisibility properties of binomial coefficients

In this paper, we aim to give full or partial proofs for the following three conjectures of V. J. W. Guo and C. Krattenthaler: (1) Let a>b be positive integers, α, β be any integers and p be a prime satisfying gcd⁡(p,a)=1. Then there exist infinitely many positive integers n for which (an+αbn+β)≡r(modp) for all integers r; (2) For any odd prime p, there are no positive integers a>b such that (anbn)≡0(modpn−1) for all n≥1; (3) For any positive integer m, there exist positive integers a and b such that am>b and (amnbn)≡0(modan−1) for all n≥1.