On monochromatic solutions of equations in groups.
On monochromatic subsets of a rectangular grid.
On multiplicative magic squares.
On polynomials with flat squares
On rainbow arithmetic progressions.
On rapid generation of .
On relatively prime sets.
On relatively prime subsets and supersets.
On repdigit polygonal numbers.
On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths
Let G be an additive finite abelian group. For every positive integer ℓ, let be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine for certain finite groups, including cyclic groups, the groups and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum subsequences...
On sets of natural numbers without solution to a noninvariant linear equation
On short zero-sum subsequences. II.
On some developments of the Erdős-Ginzburg-Ziv Theorem II
On some universal sums of generalized polygonal numbers
For m = 3,4,... those pₘ(x) = (m-2)x(x-1)/2 + x with x ∈ ℤ are called generalized m-gonal numbers. Sun (2015) studied for what values of positive integers a,b,c the sum ap₅ + bp₅ + cp₅ is universal over ℤ (i.e., any n ∈ ℕ = 0,1,2,... has the form ap₅(x) + bp₅(y) + cp₅(z) with x,y,z ∈ ℤ). We prove that p₅ + bp₅ + 3p₅ (b = 1,2,3,4,9) and p₅ + 2p₅ + 6p₅ are universal over ℤ, as conjectured by Sun. Sun also conjectured that any n ∈ ℕ can be written as and 3p₃(x) + p₅(y) + p₇(z) with x,y,z ∈ ℕ; in...
On subset sums of a fixed set
On sum-free sequences
On sumfree subsets of hypercubes
On sum-product representations in
The purpose of this paper is to investigate efficient representations of the residue classes modulo , by performing sum and product set operations starting from a given subset of . We consider the case of very small sets and composite for which not much seemed known (nontrivial results were recently obtained when is prime or when log ). Roughly speaking we show that all residue classes are obtained from a -fold sum of an -fold product set of , where and , provided the residue sets...
On sums of distinct representatives
On sum-sets and product-sets of complex numbers
We give a simple argument that for any finite set of complex numbers , the size of the the sum-set, , or the product-set, , is always large.