Intersections of binary quadratic forms in primes and the paucity phenomenon

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.