On a conjecture of Wilf about the Frobenius number

TL;DR

This paper investigates Wilf's conjecture relating the Frobenius number, the number of non-representable integers, and the number of generators, proving it under certain conditions and proposing a generalization in algebraic structures.

Contribution

The authors prove Wilf's conjecture for large, non-divisible cases with fixed ratios and introduce a broader generalization in the setting of local rings.

Findings

Wilf's conjecture holds for large a_1 not divisible by finitely many primes under fixed ratios.

The conjecture is verified for specific classes of Frobenius numbers and generators.

A new algebraic generalization of the conjecture is proposed.

Abstract

Given coprime positive integers $a_1 < ...< a_d$, the Frobenius number $F$ is the largest integer which is not representable as a non-negative integer combination of the $a_i$. Let $g$ denote the number of all non-representable positive integers: Wilf conjectured that $d \geq \frac{F+1}{F+1-g}$. We prove that for every fixed value of $\lceil \frac{a_1}{d} \rceil$ the conjecture holds for all values of $a_1$ which are sufficiently large and are not divisible by a finite set of primes. We also propose a generalization in the context of one-dimensional local rings and a question on the equality $d = \frac{F+1}{F+1-g}$.