Consecutive runs of sums of two squares

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 $a_1,a_2,a_3\mathrm{mod}q$ which are admissible in the sense that there are solutions to $x^2+y^2\equiv a_i\mathrm{mod}q$, there exist infinitely many $n$ with $E_{n+i-1}\equiv a_i\mathrm{mod}q$, for $i=1,2,3$. We also show that for any $r_1,r_2\ge 1$, there exist infinitely many $n$ with $E_{n+i-1}\equiv a_1\mathrm{mod}q$ for $1\le i\le r_1$ and $E_{n+i-1}\equiv a_2\mathrm{mod}q$ for $r_1+1\le i\le r_1+r_2$.