The Erdős–Straus conjecture is an unproven statement in number theory. The conjecture is that, for every integer that is 2 or more, there exist positive integers , , and for which In other words, the number can be written as a sum of three positive unit fractions. The conjecture is named after Paul Erdős and Ernst G. Straus, who formulated it in 1948, but it is connected to much more ancient mathematics; sums of unit fractions, like the one in this problem, are known as Egyptian fractions, because of their use in ancient Egyptian mathematics. The Erdős–Straus conjecture is one of many conjectures by Erdős, and one of many unsolved problems in mathematics concerning Diophantine equations. Although a solution is not known for all values of , infinitely many values in certain infinite arithmetic progressions have simple formulas for their solution, and skipping these known values can speed up searches for counterexamples. Additionally, these searches need only consider values of that are prime numbers, because any composite counterexample would have a smaller counterexample among its prime factors. Computer searches have verified the truth of the conjecture up to . If the conjecture is reframed to allow negative unit fractions, then it is known to be true. Generalizations of the conjecture to fractions with numerator 5 or larger have also been studied. Background and history When a rational number is expanded into a sum of unit fractions, the expansion is called an Egyptian fraction. This way of writing fractions dates to the mathematics of ancient Egypt, in which fractions were written this way instead of in the more modern vulgar fraction form with a numerator and denominator . The Egyptians produced tables of Egyptian fractions for unit fractions multiplied by two, the numbers that in modern notation would be written , such as the Rhind Mathematical Papyrus table; in these tables, most of these expansions use either two or three terms. These tables were needed, because the obvious expansion was not allowed: the Egyptians required all of the fractions in an Egyptian fraction to be different from each other. This same requirement, that all fractions be different, is sometimes imposed in the Erdős–Straus conjecture, but it makes no significant difference to the problem, because for any solution to where the unit fractions are not distinct can be converted into a solution where they are all distinct; see below. Although the Egyptians