An Extreme Family of Generalized Frobenius Numbers

TL;DR

This paper explores a broad family of generalized Frobenius numbers, revealing unusual jump patterns and establishing new results about their structure and variations, advancing understanding of this classical number theory problem.

Contribution

It introduces a specific family of parameters where generalized Frobenius numbers exhibit unnatural jumps and analyzes their properties and variations.

Findings

Generalized Frobenius numbers form arithmetic progressions.

Larger integers have exponentially many representations.

Introduces a variation of the generalized Frobenius number and proves foundational results.

Abstract

We study a generalization of the \emph{Frobenius problem}: given $k$ positive relatively prime integers, what is the largest integer $g_0$ that cannot be represented as a nonnegative integral linear combination of these parameters? More generally, what is the largest integer $g_s$ that has exactly $s$ such representations? We illustrate a family of parameters, based on a recent paper by Tripathi, whose generalized Frobenius numbers $g_0, \ g_1, \ g_2, ...$ exhibit unnatural jumps; namely, $g_0, \ g_1, \ g_k, \ g_{\binom{k+1}{k-1}}, \ g_{\binom{k+2}{k-1}}, ...$ form an arithmetic progression, and any integer larger than $g_{\binom{k+j}{k-1}}$ has at least $\binom{k+j+1}{k-}$ representations. Along the way, we introduce a variation of a generalized Frobenius number and prove some basic results about it.