Frobenius Coin Problem from Two Coins Denominations to K Coins

The Frobenius Coin Problem is a classic mathematical puzzle that asks for the largest unrepresentable integer using a given set of relatively prime coin values. Beyond finding this maximum number, exploring how the number of combination solutions grows is equally important. This paper focuses on calculating the asymptotic growth rate of these combinations, moving from the basic two-coin case to a general k-coin scenario. Instead of relying on pure algebraic estimations, which easily fail for three or more coins, this study uses a continuous complex analysis framework. The discrete counting problem is converted into generating functions, and Cauchy's Residue Theorem is applied. By analyzing the dominant highest-order pole at z=1 and using the Leibniz rule, the asymptotic growth rate for k coins is strictly determined as n k-1 /(k-1)!a 1 …a k . This analytical approach successfully overcomes the dimensional limitations of traditional algebra. Ultimately, it provides a systematic blueprint for understanding combination behaviors in broader linear Diophantine equations.