On Frobenius Numbers of Shifted Power Sequences

TL;DR

This paper provides an explicit, residue-class classified formula for the Frobenius number of shifted square sequences, resolving a longstanding open problem and confirming a conjecture from 2007.

Contribution

It introduces a novel combinatorial and number-theoretic approach to explicitly compute Frobenius numbers for shifted square sequences, confirming a previous conjecture.

Findings

Derived an explicit piecewise quadratic formula for g(A)

Confirmed a conjecture of Einstein et al. (2007)

Classified Frobenius numbers by residue classes modulo k^2

Abstract

We resolve the open problem of characterizing the Frobenius number $g(A)$ for shifted square sequences $A = (a, a+1^2, \ldots, a+k^2)$, confirming a conjecture of Einstein et al. (2007). By combining a combinatorial reduction to an optimization problem with Lagrange's Four-Square Theorem and generating function techniques, we derive an explicit formula for $g(A)$: a piecewise quadratic polynomial in $a$, classified by residue classes modulo $k^2$.