### A characterization of covering equivalence

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Let the collection of arithmetic sequences ${\{{d}_{i}n+{b}_{i}:n\in \mathbb{Z}\}}_{i\in I}$ be a disjoint covering system of the integers. We prove that if ${d}_{i}={p}^{k}{q}^{l}$ for some primes $p,q$ and integers $k,l\ge 0$, then there is a $j\ne i$ such that ${d}_{i}|{d}_{j}$. We conjecture that the divisibility result holds for all moduli.A disjoint covering system is called saturated if the sum of the reciprocals of the moduli is equal to $1$. The above conjecture holds for saturated systems with ${d}_{i}$ such that the product of its prime factors is at most $1254$.

In this paper we generalize the Pascal triangle and examine the connections among the generalized triangles and powering integers respectively polynomials. We emphasize the relationship between the new triangles and the Pascal pyramids, moreover we present connections with the binomial and multinomial theorems.

A zero-sum sequence over ℤ is a sequence with terms in ℤ that sum to 0. It is called minimal if it does not contain a proper zero-sum subsequence. Consider a minimal zero-sum sequence over ℤ with positive terms $a\u2081,...,{a}_{h}$ and negative terms $b\u2081,...,{b}_{k}$. We prove that h ≤ ⌊σ⁺/k⌋ and k ≤ ⌊σ⁺/h⌋, where $\sigma \u207a={\sum}_{i=1}^{h}{a}_{i}=-{\sum}_{j=1}^{k}{b}_{j}$. These bounds are tight and improve upon previous results. We also show a natural partial order structure on the collection of all minimal zero-sum sequences over the set i∈ ℤ : -n ≤ i ≤ n for any positive integer n.

A geometric progression of length k and integer ratio is a set of numbers of the form $a,ar,...,a{r}^{k-1}$ for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence ${\left({a}_{i}\right)}_{i=1}^{\infty}$ of positive real numbers with a₁ = 1 such that the set ${G}^{\left(k\right)}={\bigcup}_{i=1}^{\infty}({a}_{2i},{a}_{2i-1}]$ contains no geometric progression of length k and integer ratio. Moreover, ${G}^{\left(k\right)}$ is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that there is...