The Square Frobenius Number
Jonathan Chappelon (IMAG), Jorge Luis Ram\'irez Alfons\'in (IMAG, UCI)

TL;DR
This paper introduces and investigates a new variant of the Frobenius number, focusing on the largest square number not in a numerical semigroup, providing bounds, exact formulas, and conjectures for specific cases.
Contribution
It defines the square Frobenius number, derives bounds and exact formulas for certain semigroups, and proposes conjectures for broader cases.
Findings
Provided an upper bound for the square Frobenius number in arithmetic progression semigroups.
Derived exact formulas for semigroups generated by two numbers with specific differences.
Formulated conjectures for the square Frobenius number in general cases.
Abstract
Let be a numerical semigroup generated by the relatively prime positive integers . Let be an integer. In this paper, we consider the following -power variant of the Frobenius number of defined as In this paper, we investigate the case . We give an upper bound for for an infinite family of semigroups generated by {\em arithmetic progressions}. The latter turns out to be the exact value of under certain conditions. We present an exact formula for when and , study and…
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Taxonomy
TopicsCommutative Algebra and Its Applications · Rings, Modules, and Algebras · Polynomial and algebraic computation
