Consecutive runs of sums of two squares

We study the distribution of consecutive sums of two squares in arithmetic progressions. If {En}n∈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