### A cellular automaton on a torus.

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Let G be an additive abelian group of order k, and S be a sequence over G of length k+r, where 1 ≤ r ≤ k-1. We call the sum of k terms of S a k-sum. We show that if 0 is not a k-sum, then the number of k-sums is at least r+2 except for S containing only two distinct elements, in which case the number of k-sums equals r+1. This result improves the Bollobás-Leader theorem, which states that there are at least r+1 k-sums if 0 is not a k-sum.

A prime number p is called a Schenker prime if there exists n ∈ ℕ₊ such that p∤n and p|aₙ, where $a\u2099={\sum}_{j=0}^{n}(n!/j!){n}^{j}$ is a so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning p-adic valuations of aₙ when p is a Schenker prime. In particular, they conjectured that for each k ∈ ℕ₊ there exists a unique positive integer ${n}_{k}<{5}^{k}$ such that $v\u2085\left({a}_{m\xb7{5}^{k}+{n}_{k}}\right)\ge k$ for each nonnegative integer m. We prove that for every k ∈ ℕ₊ the inequality v₅(aₙ) ≥ k has exactly one solution modulo ${5}^{k}$. This confirms the...

Such problems as the search for Wieferich primes or Wall-Sun-Sun primes are intensively studied and often discused at present. This paper is devoted to a similar problem related to the Tribonacci numbers.