Positive density for consecutive runs of sums of two squares

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Noam Kimmel and Vivian Kuperberg, Positive density for consecutive runs of sums of two squares. The selected text is arXiv:2406.04174v2, revised 26 March 2025. The source PDF is available from arXiv.

The arXiv record, read 2026-10-15, specifies arXiv's non-exclusive distribution license. This does not grant redistribution under an open license. No file of this source is held.

Scope

Let $E_n$ list the sums of two integer squares in increasing order. For an odd squarefree modulus $q$, two reduced residue classes modulo $q$, and two fixed positive run lengths, the main theorem gives a positive proportion of starting values $E_n$ for which the consecutive terms first have the first residue and then the second. The modulus and the residue hypotheses belong to the theorem. This is a result about sums of two squares, not a theorem about prime runs.

The introduction compares this result with Maynard's positive-density theorem for a constant residue run of consecutive primes. It also describes the much weaker state of knowledge for prescribed nonconstant prime patterns in the selected version.

Read status: claims checked. The abstract, introduction and main theorem were read in the arXiv TeX source at statement depth. Neither its sieve proof nor the proofs of the prime results it cites have been checked here.

Bears on. Problem 251: the introductory comparison helps delimit proposed prime-pattern inputs. No transfer of the sums-of-two-squares theorem to the sequence of primes is used.