  We study the distribution of consecutive sums of two squares in arithmetic
progressions. We show that for any odd squarefree modulus $q$, any two reduced
congruence classes $a_1$ and $a_2$ mod $q$, and any $r_1,r_2 \ge 1$, a positive
density of sums of two squares begin a chain of $r_1$ consecutive sums of two
squares, all of which are $a_1$ mod $q$, followed immediately by a chain of
$r_2$ consecutive sums of two squares, all of which are $a_2$ mod $q$. This is
an analog of the result of Maynard for the sequence of primes, showing that for
any reduced congruence class $a$ mod $q$ and for any $r \ge 1$, a positive
density of primes begin a sequence of $r$ consecutive primes, all of which are
$a$ mod $q$.
