Abstract
The number of solutions to a2+b2=c2+d2≤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.
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