Abstract
Given coprime positive integers \(a_1 < \cdots < a_d\), 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 \(d = \frac{F+1}{F+1-g}\).
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Moscariello, A., Sammartano, A. On a conjecture by Wilf about the Frobenius number. Math. Z. 280, 47–53 (2015). https://doi.org/10.1007/s00209-015-1412-0
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DOI: https://doi.org/10.1007/s00209-015-1412-0
Keywords
- Diophantine Frobenius problem
- Coin problem
- Wilf’s conjecture
- Numerical semigroup
- Apéry set
- Length inequality
- One-dimensional local ring