Let p be any prime, and let α and n be nonnegative integers. Let r ∈ Z and f (x) ∈ Z[x]. We establish the congruence p deg f k≡r (mod p α) n k (−1) k f k − r p α ≡ 0 mod p ∞ i=α ⌊n/p i ⌋. This result is a vast generalization of a conjecture arising from algebraic topology. An application to Bernoulli polynomials is also given.

The AKS conjecture states that (ζ − 1) ≡ ζ − 1 (mod n) for ζ an r-th root of unity with r prime, only if n is a prime and r n − 1. If n is a prime, n-th power map would be a Frobenius automorphism of Fn(ζ), and thus the congruence will hold true. In general, n-th power map is not a field automorphism, but under suitable conditions, it can be a group automorphism of the multiplicative group of c...

Let p be an odd prime and a be a positive integer. We show that p−1 k=0 (−1) (a−1)k p − 1 k a ≡ 2 a(p−1) + a(a − 1)(3a − 4) 48 p 3 B p−3 (mod p 4), which is a generalization of a congruence due to Morley.

A metabelian group G of order 1440 is constructed which provides a counterexample to a conjecture of Zassenhaus on automorphisms of integral group rings. The group is constructed in the spirit of [8]. An augmented automorphism of ZG which has no Zassenhaus factorization is given explicitly (this was already done in [7] for a group of order 6720), but this time only a few distinguished group rin...

Let R be a field of any characteristic and A a principal ideal domain. We make a conjecture that asserts that any Hecke operator T acts punctually on any Hecke eigenclass in the stable homology with trivial coefficients R of a principal congruence subgroup Γ in GL(n, A), i.e. as multiplication by the number of single cosets contained in T . In the case where A = Z, this conjecture implies that ...

For distributive double p-algebras, a close connection is established between being congruence coherent and congruence regular. Every congruence regular distributive double p-algebra is congruence coherent. Even though every congruence coherent distributive double p-algebra that has either a non-empty core or finite range is congruence regular, an example is given that is congruence coherent bu...