Sums of squares of integers from residue classes

2025/08/11 by Kim, Daejun
#11E12 #11E20 #11E25 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.08106

Abstract

A subset A⊆ℤ is called s-almost square universal if every sufficiently large positive integer can be written as a sum of at most s squares of integers from A. In this article, we study the minimal number ASU(A_d,m) with this property, where A_d,m denotes the residue class of d modulo m, with m∈ℕ and d∈ℤ. We further prove that A_d,m is s-square universal for some s∈ℕ if and only if d ≡ ± 1 \pmodm, and determine the minimal such number SU(A_d,m) in these cases.

Citations

• Sums of squares of integers except for a fixed one
• Theta series of ternary quadratic lattice cosets
• On a quadratic Waring's problem with congruence conditions
• A sum of squares not divisible by a prime
• Almost universal ternary sums of polygonal numbers
• Strong approximation for certain quadric fibrations with compact fibers
• Representations of integral quadratic polynomials
• Quaternary Quadratic Forms Representing all Integers

Related

• On the sums of squares of exceptional units in residue class rings
• Universal Sums of Triangular Numbers and Squares
• Finiteness theorems for universal sums of squares of almost primes
• Representing Integers as the Sum of Two Squares in the Ring \Z_n
• A sum of squares not divisible by a prime