Related Expansions
In general, if one wants an Egyptian fraction expansion in which the denominators are constrained in some way, it is possible to define a greedy algorithm in which at each step one chooses the expansion
where d is chosen, among all possible values satisfying the constraints, as small as possible such that xd > y and such that d is distinct from all previously chosen denominators. For instance, the Engel expansion can be viewed as an algorithm of this type in which each successive denominator must be a multiple of the previous one. However, it may be difficult to determine whether an algorithm of this type can always succeed in finding a finite expansion. In particular, the odd greedy expansion of a fraction x/y is formed by a greedy algorithm of this type in which all denominators are constrained to be odd numbers; it is known that, whenever y is odd, there is a finite Egyptian fraction expansion in which all denominators are odd, but it is not known whether the odd greedy expansion is always finite.
Read more about this topic: Greedy Algorithm For Egyptian Fractions
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