1 modulo 4, so the searches only need to test values that are 1 modulo 4. One way to make progress on this problem is to collect more modular identities, allowing computer searches to reach higher limits with fewer tests. The number of distinct solutions to the problem, as a function of , has also been found by computer searches for small and appears to grow somewhat irregularly with . Starting with , the numbers of distinct solutions with distinct denominators are Even for larger there can sometimes be relatively few solutions; for instance there are only seven distinct solutions for . Theoretical results In the form , a polynomial equation with integer variables, the Erdős–Straus conjecture is an example of a Diophantine equation. The Hasse principle for Diophantine equations suggests that these equations should be studied using modular arithmetic. If an polynomial equation has a solution in the integers, then taking this solution modulo , for any integer , provides a solution in modulo- arithmetic. In the other direction, if an equation has a solution modulo for every prime power , then in some cases it is possible to piece together these modular solutions, using methods related to the Chinese remainder theorem, to get a solution in the integers. The power of the Hasse principle to solve some problems is limited by the Manin obstruction, but for the Erdős–Straus conjecture this obstruction does not exist. On the face of it this principle makes little sense for the Erdős–Straus conjecture. For every , the equation is easily solvable modulo any prime, or prime power, but there appears to be no way to piece those solutions together to get a positive integer solution to the equation. Nevertheless, modular arithmetic, and identities based on modular arithmetic, have proven a very important tool in the study of the conjecture. Modular identities For values of satisfying certain congruence relations, one can find an expansion for automatically as an instance of a polynomial identity. For instance, whenever is 2 modulo 3, has the expansion Here each of the three denominators , , and is a polynomial of , and each is an integer whenever is 2 modulo 3. The greedy algorithm for Egyptian fractions finds a solution in three or fewer terms whenever is not 1 or 17 mod 24, and the 17 mod 24 case is covered by the 2 mod 3 relation, so