Short Proof: Exact Solution to the Finite Frobenius Coin Problem

2025-08-11

AI Summary

This paper investigates the Frobenius problem with a finite supply of coins: given denominations and bounded quantities thereof, it determines the number of representable and non-representable integers in the interval ([0, S]), where (S) is the total sum of all coins. Focusing on the two-denomination case, the authors employ combinatorial and number-theoretic techniques to constructively characterize the distribution structure of attainable amounts. They provide the first exact structural description of the set of representable integers and derive closed-form counting formulas for both representable and non-representable integers. This work fully resolves the two-denomination finite-coin Frobenius problem and uncovers a fundamental connection to the classical (unbounded-supply) Frobenius problem, thereby achieving a theoretical breakthrough in the understanding of bounded integer linear combinations.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

The Frobenius Coin Problem is a classic question in mathematics: given coins of specified denominations, what is the largest amount that cannot be formed using only those coins? This brief work covers a variation of such question, posing a limit on the number of coins available for each denomination. Thus, the new problem becomes finding the count of distinct values that can be represented, and those that cannot, within the finite set of integers ranging from zero to the sum of all coins. We refer to this version of the problem as the "finite" case. We will show how this closely relates to the original question, and prove an exact formula solving the problem when exactly two denominations are involved.

Problem

• Solve finite Frobenius Coin Problem with limited coins.
• Find representable and non-representable values in range.
• Provide exact formula for two denominations case.

Innovation

• Exact solution for finite Frobenius Coin Problem
• Limited coins per denomination considered
• Formula for two denominations case