exist positive integers , , and such that For instance, for , there are two solutions: Multiplying both sides of the equation by leads to an equivalent polynomial form for the problem. Distinct unit fractions Some researchers additionally require that the integers , , and be distinct from each other, as the Egyptians would have, while others allow them to be equal. For , it does not matter whether they are required to be distinct: if there exists a solution with any three integers, then there exists a solution with distinct integers. This is because two identical unit fractions can be replaced through one of the following two expansions: (according to whether the repeated fraction has an even or odd denominator) and this replacement can be repeated until no duplicate fractions remain. For , however, the only solutions are permutations of . Negative-number solutions The Erdős–Straus conjecture requires that all three of , , and be positive. This requirement is essential to the difficulty of the problem. Even without this relaxation, the Erdős–Straus conjecture is difficult only for odd values of , and if negative values were allowed then the problem could be solved for every odd by the following formula: Computational results If the conjecture is false, it could be proven false simply by finding a number that has no three-term representation. In order to check this, various authors have performed brute-force searches for counterexamples to the conjecture. Searches of this type have confirmed that the conjecture is true for all up to . In such searches, it is only necessary to look for expansions for numbers where is a prime number. This is because, whenever has a three-term expansion, so does for all positive integers . To find a solution for , just divide all of the unit fractions in the solution for by : If were a counterexample to the conjecture, for a composite number , every prime factor of would also provide a counterexample that would have been found earlier by the brute-force search. Therefore, checking the existence of a solution for composite numbers is redundant, and can be skipped by the search. Additionally, the known modular identities for the conjecture (see below) can speed these searches by skipping over other values known to have a solution. For instance, the greedy algorithm finds an expansion with three or fewer terms for every number where is not