Intersections of binary quadratic forms in primes and the paucity phenomenon

TL;DR

This paper investigates the distribution of solutions to binary quadratic form equations with prime variables, revealing the scarcity of off-diagonal solutions and extending previous results to new prime configurations.

Contribution

It extends the analysis of solutions to quadratic forms with prime variables to new cases, combining multiple advanced techniques.

Findings

Diagonal solutions dominate in prime-variable quadratic forms

Off-diagonal solutions are scarce, confirming the paucity phenomenon

New cases with different prime restrictions are analyzed successfully

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.