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.
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.
, it is unknown whether four terms are sometimes needed, or whether it is possible to express all fractions of the form
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
[1] 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,[1] but sometimes it is smaller.
[3] 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
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
, and if negative values were allowed then the problem could be solved for every odd
In order to check this, various authors have performed brute-force searches for counterexamples to the conjecture.
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
One way to make progress on this problem is to collect more modular identities, allowing computer searches to reach higher limits with fewer tests.
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.
The greedy algorithm for Egyptian fractions finds a solution in three or fewer terms whenever
for which these two methods do not find expansions in three or fewer terms are those congruent to 1 mod 24.
[12] Polynomial identities listed by Mordell (1967) provide three-term Egyptian fractions for
By combining larger classes of modular identities, Webb and others showed that the natural density of potential counterexamples to the conjecture is zero: as a parameter
[13] If it were possible to find solutions such as the ones above for enough different moduli, forming a complete covering system of congruences, the problem would be solved.
However, as Mordell (1967) showed, a polynomial identity that provides a solution for values of
For instance, 2 is a non-square mod 3, so Mordell's result allows the existence of an identity for
[14] Despite Mordell's result limiting the form of modular identities for this problem, there is still some hope of using modular identities to prove the Erdős–Straus conjecture.
One possible approach to proving the conjecture would be to find for each prime
[12] Elsholtz & Tao (2013) showed that the average number of solutions to the
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.
[17] The generalized version of the conjecture is equivalent to the statement that the number of unexpandable fractions is not just sublinear but bounded.