Consecutive runs of sums of two squares

We study the distribution of consecutive sums of two squares in arithmetic progressions. If {En}n is an element of N is the sequence of sums of two squares in increasing order, we show that for any modulus q and any congruence classes a1, a2, a3 mod q which are admissible in the sense that there are solutions to x2 + y2 ai mod q, there exist infinitely many n with En+i-1 ai mod q, for i = 1, 2, 3. We also show that for any r1, r2 > 1, there exist infinitely many n with En+i-1 a1 mod q for 1 < i < r1 and En+i-1 a2 mod q for r1 + 1 < i < r1 + r2. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.