Positive density for consecutive runs of sums of two squares

article Positive density for consecutive runs of sums of two squares published yes Noam Kimmel author Vivian Zieve Kuperberg author c3bac823-112d-11f0-a3f5-c264f852e697 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 a1 and a2 mod q, and any r1,r2 ≥ 1, a positive density of sums of two squares begin a chain of r1 consecutive sums of two squares, all of which are a1 mod q, followed immediately by a chain of r2 consecutive sums of two squares, all of which are a2 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 ≥ 1, a positive density of primes begin a sequence of r consecutive primes, all of which are a mod q Cambridge University Press2025 eng Journal of the Institute of Mathematics of Jussieu 1474-7480 1475-3030 2406.0417410.1017/s1474748025000131 2451995-2046 yes Kimmel N, Kuperberg VZ. Positive density for consecutive runs of sums of two squares. Journal of the Institute of Mathematics of Jussieu. 2025;24(5):1995-2046. doi:<a href="https://doi.org/10.1017/s1474748025000131">10.1017/s1474748025000131</a> N. Kimmel, V.Z. Kuperberg, Journal of the Institute of Mathematics of Jussieu 24 (2025) 1995–2046. N. Kimmel and V. Z. Kuperberg, “Positive density for consecutive runs of sums of two squares,” Journal of the Institute of Mathematics of Jussieu, vol. 24, no. 5. Cambridge University Press, pp. 1995–2046, 2025. Kimmel N, Kuperberg VZ. 2025. Positive density for consecutive runs of sums of two squares. Journal of the Institute of Mathematics of Jussieu. 24(5), 1995–2046. Kimmel, Noam, and Vivian Zieve Kuperberg. “Positive Density for Consecutive Runs of Sums of Two Squares.” Journal of the Institute of Mathematics of Jussieu. Cambridge University Press, 2025. <a href="https://doi.org/10.1017/s1474748025000131">https://doi.org/10.1017/s1474748025000131</a>. Kimmel, N., & Kuperberg, V. Z. (2025). Positive density for consecutive runs of sums of two squares. Journal of the Institute of Mathematics of Jussieu. Cambridge University Press. <a href="https://doi.org/10.1017/s1474748025000131">https://doi.org/10.1017/s1474748025000131</a> Kimmel, Noam, and Vivian Zieve Kuperberg. “Positive Density for Consecutive Runs of Sums of Two Squares.” Journal of the Institute of Mathematics of Jussieu, vol. 24, no. 5, Cambridge University Press, 2025, pp. 1995–2046, doi:<a href="https://doi.org/10.1017/s1474748025000131">10.1017/s1474748025000131</a>. 222042026-06-29T13:01:08Z2026-07-14T11:50:36Z