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 Erdős [4] 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 [3] 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.

解决方案的数量${\mathrm{一种}}^2+b^2=C^2+d^2\le X$in integers 是一个众所周知的结果，而如果将所有变量限制为素数 Erdős [4] 表明只有对角线解，即具有$\{\mathrm{一种},b\}=\{C,d\}$对主要项有贡献，因此缺乏非对角线解。Daniel [3] 考虑了以下情况$\mathrm{一种},C$是素数，并证明主项既有对角线贡献，也有非对角线贡献。在这里我们研究剩下的情况，即只有c是素数并且两者都是$C,d$是素数，最后，当$b,C,d$通过结合 Daniel、Hooley 和 Plaksin 的技术，是素数。