Consecutive runs of sums of two squares

TL;DR

This paper investigates the distribution of consecutive sums of two squares within arithmetic progressions, proving the existence of infinitely many such sequences matching given residue classes under certain conditions.

Contribution

It establishes the infinite occurrence of consecutive sums of two squares in specified residue classes for any admissible moduli and classes, extending understanding of their distribution.

Findings

Infinitely many sequences of three consecutive sums of two squares in specified residue classes.

Existence of infinitely many sequences with prescribed patterns of sums of two squares modulo q.

Distribution results for sums of two squares in arithmetic progressions.

Abstract

We study the distribution of consecutive sums of two squares in arithmetic progressions. If $\{E_n\}_{n \in \mathbb{N}}$ is the sequence of sums of two squares in increasing order, we show that for any modulus $q$ and any congruence classes $a_1,a_2,a_3 \mod q$ which are admissible in the sense that there are solutions to $x^2 + y^2 \equiv a_i \mod q$, there exist infinitely many $n$ with $E_{n+i-1} \equiv a_i \mod q$, for $i = 1,2,3$. We also show that for any $r_1, r_2 \ge 1$, there exist infinitely many $n$ with $E_{n+i-1} \equiv a_1 \mod q$ for $1 \le i \le r_1$ and $E_{n+ i - 1} \equiv a_2 \mod q$ for $r_1 + 1 \le i \le r_1 + r_2$.