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

Keywords