In this paper we provide bounds for the size of the solutions of the Diophantine equation x(x+ 1)(x+ 2)(x+ 3)(x+ k)(x+ k+ 1)(x+ k+ 2)(x+ k+ 3) = y, where 4 ≤ k ∈ N is a parameter. We also determine all integral solutions for 1 ≤ k ≤ 10.

Let G be a finite Abelian group of order |G| = n, and let S = g1 · . . . · gn−1 be a sequence over G such that all nonempty zero-sum subsequences of S have the same length. In this paper, we completely determine the structure of these sequences.

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. In doing so, we make fundamental connections with both discrete geometry and algebraic combinatorics. particular, show are necessarily optimal packings in projective space, introduce fruitful generalization regular abelian distance-regular antipodal covers complete gr...