The pentagonal theorem of sixty-three and generalizations of Cauchy's lemma

TL;DR

This paper proves a new theorem characterizing which integers can be expressed as sums of pentagonal numbers, and introduces a method to generalize Cauchy's lemma using quadratic form representations.

Contribution

It establishes the 'pentagonal theorem of 63' and develops a novel approach to generalize Cauchy's lemma via quadratic form representations.

Findings

A complete characterization of integers representable as sums of pentagonal numbers.

Introduction of a method to generalize Cauchy's lemma using quadratic forms.

Proof that the specified set of integers is necessary and sufficient for representation.

Abstract

In this article, we study the representability of integers as sums of pentagonal numbers, where a pentagonal number is an integer of the form $P_5(x)=\frac{3x^2-x}{2}$ for some non-negative integer $x$. In particular, we prove the "pentagonal theorem of $63$", which states that a sum of pentagonal numbers represents every non-negative integer if and only if it represents the integers $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $11$, $13$, $14$, $17$, $18$, $19$, $23$, $28$, $31$, $33$, $34$, $39$, $42$, and $63$. We also introduce a method to obtain a generalized version of Cauchy's lemma using representations of binary integral quadratic forms by quaternary quadratic forms, which plays a crucial role in proving the results.