did not always find expansions using as few terms as possible, later mathematicians have been interested in the question of how few terms are needed. Every fraction has an expansion of at most terms, so in particular needs at most two terms, needs at most three terms, and needs at most four terms. For , two terms are always needed, and for , three terms are sometimes needed, so for both of these numerators, the maximum number of terms that might be needed is known. However, for , it is unknown whether four terms are sometimes needed, or whether it is possible to express all fractions of the form using only three unit fractions; this is the Erdős–Straus conjecture. Thus, the conjecture covers the first unknown case of a more general question, the problem of finding for all the maximum number of terms needed in expansions for fractions . One way to find short (but not always shortest) expansions uses the greedy algorithm for Egyptian fractions, first described in 1202 by Fibonacci in his book Liber Abaci. This method chooses one unit fraction at a time, at each step choosing the largest possible unit fraction that would not cause the expanded sum to exceed the target number. After each step, the numerator of the fraction that still remains to be expanded decreases, so the total number of steps can never exceed the starting numerator, but sometimes it is smaller. For example, when it is applied to , the greedy algorithm will use two terms whenever is 2 modulo 3, but there exists a two-term expansion whenever has a factor that is 2 modulo 3, a weaker condition. For numbers of the form , the greedy algorithm will produce a four-term expansion whenever is 1 modulo 4, and an expansion with fewer terms otherwise. Thus, another way of rephrasing the Erdős–Straus conjecture asks whether there exists another method for producing Egyptian fractions, using a smaller maximum number of terms for the numbers . The Erdős–Straus conjecture was formulated in 1948 by Paul Erdős and Ernst G. Straus, and published by . Richard Obláth also published an early work on the conjecture, a paper written in 1948 and published in 1950, in which he extended earlier calculations of Straus and Harold N. Shapiro in order to verify the conjecture for all . Formulation The conjecture states that, for every integer , there