### Consecutive sums of two squares in arithmetic progressions

Heilbronn Number Theory Seminar

17th April 2024, 4:00 pm – 5:00 pm

Fry Building, 2.04

There are infinitely many primes whose last digit is 1 such that the next prime also ends in a 1, and in fact these primes have positive density in the set of all primes. However, it is an open problem to show that there are infinitely many primes ending in 1 such that the next prime ends in 3. In this talk, we'll instead consider the sequence of sums of two squares in increasing order. We'll show that there are infinitely many sums of two squares ending in 1 such that the next sum of two squares ends in 3, and in fact that these sums of two squares have positive density in the set of all sums of two squares. Joint work with Noam Kimmel.

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