Positive density for consecutive runs of sums of two squares Kimmel, Noam Kuperberg, Vivian Zieve 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 Press 2025 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/22204 Kimmel N, Kuperberg VZ. Positive density for consecutive runs of sums of two squares. <i>Journal of the Institute of Mathematics of Jussieu</i>. 2025;24(5):1995-2046. doi:<a href="https://doi.org/10.1017/s1474748025000131">10.1017/s1474748025000131</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1017/s1474748025000131 info:eu-repo/semantics/altIdentifier/issn/1474-7480 info:eu-repo/semantics/altIdentifier/e-issn/1475-3030 info:eu-repo/semantics/altIdentifier/arxiv/2406.04174 info:eu-repo/semantics/openAccess