A note on the $Σ_2^P$-completeness of the Frobenius number

2026-08-30

AI Summary

该文解决了验证Frobenius数是否大于等于给定值k的问题，通过提供基于Matsubara(2016)的证明来展示其为Σ2P-完全问题。

Abstract

Given a finite set $A$ of natural numbers whose greatest common divisor is one, the Frobenius number $g(A)$ is the largest integer that is not a non-negative integer combination of the numbers in $A$. In a 2016 preprint, Matsubara states that given $A$ and $k$, deciding if $g(A) \geq k$ is $Σ_2^P$-complete. A decade has passed since without peer-reviewed publication of this result. At the same time, the community has found it difficult to verify this result. In this note, we give a write-up of the completeness proof based on Matsubara (2016).

Problem

• Frobenius number
• Σ2P-completeness
• natural numbers

Innovation

• Σ₂^P-completeness
• Frobenius number
• proof verification