Sums of squares of integers except for a fixed one

TL;DR

This work investigates sums of squares of integers excluding a fixed integer $\rho$ by introducing $S_\rho$, $\Sigma_\rho(n)$, $k_\rho(n)$, $I(\rho)$, and $M(\rho)$. It proves a universal bound: every sufficiently large $n$ is a sum of at most four squares from $S_\rho$, with the threshold $n \ge 550\rho^{2}$. It then provides a complete classification: $M(\rho)=4$ whenever $\rho$ has a prime factor greater than $3$ or is divisible by $9$, and $M(\rho)=5$ in the remaining cases, detailing exceptional sets $N(\rho)$ for special forms like $\rho=2^{a+1}$ and $\rho=3\cdot 2^{a+1}$ and employing primitive representations over $\mathbb Z_3$ to control decompositions. Overall, the results extend Dubouis's classical findings on sums of nonzero squares to the setting where a fixed summand is excluded, mapping the landscape of restricted-square representations in $\mathbb Z$.

Abstract

In this article, we study a sum of squares of integers except for a fixed one. For any nonnegative integer $n$, we find the minimum number of squares of integers except for $n$ whose sums represent all positive integers that are represented by a sum of squares except for it. This problem could be considered as a generalization of Dubouis's result for the case when $n=0$.

Theorems & Definitions (16)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof