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arXiv · 2406.04174

Positive density for consecutive runs of sums of two squares

Noam Kimmel·Vivian Kuperberg
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Abstract

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 $q$.

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Noam Kimmel, Vivian Kuperberg. 2024-06-06. Positive density for consecutive runs of sums of two squares. https://doi.org/10.1017/s1474748025000131

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