On a conjecture of Wilf about the Frobenius number

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 $d\ge \frac{F+1}{F+1-g}$. 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 $d=\frac{F+1}{F+1-g}$.

References in corpus (1)

• Fibonacci-Like Behavior of the Number of Numerical Semigroups of a Given Genus

Cited by in corpus (1)

• Eliahou number, Wilf function and concentration of a numerical semigroup