a counterexample to the conjecture and the conjecture would be true. The number of solutions showed that the average number of solutions to the problem (averaged over the prime numbers up to ) is upper bounded polylogarithmically in . For some other Diophantine problems, the existence of a solution can be demonstrated through asymptotic lower bounds on the number of solutions, but this works best when the number of solutions grows at least polynomially, so the slower growth rate of Elsholtz and Tao's result makes a proof of this type less likely. Elsholtz and Tao classify solutions according to whether one or two of , , or is divisible by ; for prime , these are the only possibilities, although (on average) most solutions for composite are of other types. Their proof uses the Bombieri–Vinogradov theorem, the Brun–Titchmarsh theorem, and a system of modular identities, valid when is congruent to or modulo , where and are any two coprime positive integers and is any odd factor of . For instance, setting gives one of Mordell's identities, valid when is 3 mod 4. Generalizations As with fractions of the form , it has been conjectured that every fraction (for ) can be expressed as a sum of three positive unit fractions. A generalized version of the conjecture states that, for any positive , all but finitely many fractions can be expressed as a sum of three positive unit fractions. The conjecture for fractions was made by Wacław Sierpiński in a 1956 paper, which went on to credit the full conjecture to Sierpiński's student Andrzej Schinzel. Even if the generalized conjecture is false for any fixed value of , then the number of fractions with in the range from 1 to that do not have three-term expansions must grow only sublinearly as a function of . In particular, if the Erdős–Straus conjecture itself (the case ) is false, then the number of counterexamples grows only sublinearly. Even more strongly, for any fixed , only a sublinear number of values of need more than two terms in their Egyptian fraction expansions. The generalized version of the conjecture is equivalent to the statement that the number of unexpandable fractions is not just sublinear but bounded. When is an odd number, by analogy to the problem of odd greedy expansions for Egyptian fractions, one may ask for solutions to in which , , and are