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arXiv · 2107.05525

Intersections of binary quadratic forms in primes and the paucity phenomenon

Alisa Sedunova
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Abstract

The number of solutions to $a^2+b^2=c^2+d^2 \le x$ in integers is a well-known result, while if one restricts all the variables to primes Erdos showed that only the diagonal solutions, namely, the ones with $\{a,b\}=\{c,d\}$ contribute to the main term, hence there is a paucity of the off-diagonal solutions. Daniel considered the case of $a,c$ being prime and proved that the main term has both the diagonal and the non-diagonal contributions. Here we investigate the remaining cases, namely when only $c$ is a prime and when both c,d are primes and, finally, when $b,c,d$ are primes by combining techniques of Daniel, Hooley and Plaksin.

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Alisa Sedunova. 2021-07-12. Intersections of binary quadratic forms in primes and the paucity phenomenon. https://doi.org/10.1016/j.jnt.2021.06.035

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