相关论文: On product representations of squares

A natural generalization of base B expansions is Zeckendorf's Theorem: every integer can be uniquely written as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, with $F_{n+1} = F_n + F_{n-1}$ and $F_1=1, F_2=2$. If instead we allow the…

Let $K$ be a Birch field, that is, a field for which every diagonal form of odd degree in sufficiently many variables admits a non-zero solution; for example, $K$ could be the field of rational numbers. Let $f_1, \ldots, f_r$ be homogeneous…

In this paper, as a main theorem, we prove that the decision version of the Frobenius problem is Sigma_2^P-complete under Karp reductions.Given a finite set A of coprime positive integers, we call the greatest integer that cannot be…

A family $\mathcal{F}\subset 2^G$ of subsets of an abelian group $G$ is a Sidon system if the sumsets $A+B$ with $A,B\in \mathcal{F}$ are pairwise distinct. Cilleruelo, Serra and the author previously proved that the maximum size $F_k(n)$…

We show that for every positive integer $k$ there are positive constants $C$ and $c$ such that if $A$ is a subset of $\{1, 2, \dots, n\}$ of size at least $C n^{1/k}$, then, for some $d \leq k-1$, the set of subset sums of $A$ contains a…

Let $F_p$ be the field of a prime order $p.$ For a subset $A\subset F_p$ we consider the product set $A(A+1).$ This set is an image of $A\times A$ under the polynomial mapping $f(x,y)=xy+x:F_p\times F_p\to F_p.$ In the present paper we show…

Let $f(n)$ be a Steinhaus random multiplicative function. Let $A\subset [1, N]$ be a finite set of integers. We show that \[\frac{1}{\sqrt{|A|}} \sum_{n\in A} f(n) \xrightarrow[]{d} \mathcal{CN}(0,1)\] forces that $|A|=o(N)$. We prove that…

Let $r_k(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ squares. We rigorously prove for the first time a Voronoi summation formula for $r_k(n), k\geq2,$ proved incorrectly by A. I. Popov and later…

Zeckendorf proved that every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers. Once this has been shown, it's natural to ask how many summands are needed. Using a continued fraction approach,…

The product dimension of a graph $G$ is the minimum possible number of proper vertex colorings of $G$ so that for every pair $u,v$ of non-adjacent vertices there is at least one coloring in which $u$ and $v$ have the same color. What is the…

We explore variants of Erd\H os' unit distance problem concerning dot products between successive pairs of points chosen from a large finite subset of either $\mathbb F_q^d$ or $\mathbb Z_q^d,$ where $q$ is a power of an odd prime.…

We say that the families $\mathcal F_1,\ldots, \mathcal F_{s+1}$ of $k$-element subsets of $[n]$ are cross-dependent if there are no pairwise disjoint sets $F_1,\ldots, F_{s+1}$, where $F_i\in \mathcal F_i$ for each $i$. The rainbow version…

Let $F\in\mathbb{C}[x,y,s,t]$ be an irreducible constant-degree polynomial, and let $A,B,C,D\subset\mathbb{C}$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{8/3})$ points of the Cartesian product $A\times B\times…

Let $\mathbf{k} := (k_1,\dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges…

Recently, Andrews and Bachraoui considered a generating function $F_{k,m}(q)$ associated with certain two-color partitions, and conjectured that this function has non-negative coefficients for $m=1$. They showed this property for $1 \leq k…

Let $F_{n}$ and $L_n$ be the $n$th Fibonacci and Lucas number, respectively. For each positive integer $m$, the order of appearance of $m$ in the Fibonacci sequence, denoted by $z(m)$, is the smallest positive integer $k$ such that $m$…

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer…

A family of sets is said to be \emph{intersecting} if any two sets in the family have nonempty intersection. In 1973, Erd\H{o}s raised the problem of determining the maximum possible size of a union of $r$ different intersecting families of…

We consider the problem of finding sum of squares (sos) expressions to establish the non-negativity of a symmetric polynomial over a discrete hypercube whose coordinates are indexed by the $k$-element subsets of $[n]$. For simplicity, we…

We prove that a subset $A\subseteq [1, N]$ with \[\sum_{n\in A}\frac{1}{n} \ge (\log N)^{4/5 + o(1)}\] contains a subset $B$ such that \[\sum_{n\in B} \frac{1}{n} = 1.\] Our techniques refine those of Croot and of Bloom. Using our…