On Frobenius problem with restrictions on common divisors of coefficients

TL;DR

This paper extends the Frobenius problem by exploring representations of large integers with restrictions on the divisibility of coefficients by m-th powers, considering common divisors among subsets of these coefficients.

Contribution

It proves new results on the representation of large integers with specific divisibility restrictions on the coefficients, generalizing classical Frobenius problem conditions.

Findings

Every sufficiently large integer can be expressed with coefficients having no common m-th power divisor, but subsets of size t share such divisors.

Every large integer can be written as a sum of integers with no common m-th power divisor, but subsets of size s-1 share such divisors.

The results hold under conditions on the parameters m, s, t, and the gcd of the initial sequence.

Abstract

Let $m,s,t$ are positive integers with $t\leq s-2$ and $a_1,a_2,\ldots,a_s$ are positive integers such that $(a_1,a_2,\ldots,a_{s-1})=1$. In the paper we prove that every sufficiently large positive integer can be written in the form $a_1\mu_1+a_2\mu_2+\ldots+a_s\mu_m$, where positive integers $\mu_1,\mu_2,\ldots,\mu_s$ have no common divisor being $m$-th power of a positive integer greater than $1$ but each $t$ of the values of $\mu_1,\mu_2,\ldots,\mu_n$ have a common divisor being $m$-th power of a positive integer greater than $1$. Moreover, we show that every sufficiently large positive integer can be written as a sum of positive integers $\mu_1,\mu_2,\ldots,\mu_n$ with no common divisor being $m$-th power of a positive integer greater than $1$ but each $s-1$ of the values of $\mu_1,\mu_2,\ldots,\mu_s$ have a common divisor being $m$-th power of a positive integer greater than $1$.