On a conjecture of Wilf about the Frobenius number

Alessio Moscariello, Alessio Sammartano

Published 2014-08-22Version 1

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 n denote the number of integers less than F admitting such a representation: Wilf conjectured that F < nd. We prove that for every fixed value of floor(a_1/d) the conjecture holds for all values of a1 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 F + 1 = nd.