LAUSR

Abstract. The number of solutions to a+b = c+d ≤ 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 nondiagonal 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.