Egyptian Fraction - Open Problems

Open Problems

For more information on this subject, see odd greedy expansion and Erdős–Straus conjecture.

Some notable problems remain unsolved with regard to Egyptian fractions, despite considerable effort by mathematicians.

• The Erdős–Straus conjecture concerns the length of the shortest expansion for a fraction of the form 4/n. Does an expansion

exist for every n? It is known to be true for all n < 1014, and for all but a vanishingly small fraction of possible values of n, but the general truth of the conjecture remains unknown.

• It is unknown whether an odd greedy expansion exists for every fraction with an odd denominator. If Fibonacci's greedy method is modified so that it always chooses the smallest possible odd denominator, under what conditions does this modified algorithm produce a finite expansion? An obvious necessary condition is that the starting fraction x/y have an odd denominator y, and it is conjectured but not known that this is also a sufficient condition. It is known (Breusch 1954; Stewart 1954) that every x/y with odd y has an expansion into distinct odd unit fractions, constructed using a different method than the greedy algorithm.
• It is possible to use brute-force search algorithms to find the Egyptian fraction representation of a given number with the fewest possible terms (Stewart 1992) or minimizing the largest denominator; however, such algorithms can be quite inefficient. The existence of polynomial time algorithms for these problems, or more generally the computational complexity of such problems, remains unknown.

Guy (2004) describes these problems in more detail and lists numerous additional open problems.