Sylvester's Sequence and Closest Approximation
Sylvester's sequence 2, 3, 7, 43, 1807, ... can be viewed as generated by an infinite greedy expansion of this type for the number one, where at each step we choose the denominator instead of . Truncating this sequence to k terms and forming the corresponding Egyptian fraction, e.g. (for k = 4)
results in the closest possible underestimate of 1 by any k-term Egyptian fraction (Curtiss 1922; Soundararajan 2005). That is, for example, any Egyptian fraction for a number in the open interval (1805/1806,1) requires at least five terms. Curtiss (1922) describes an application of these closest-approximation results in lower-bounding the number of divisors of a perfect number, while Stong (1983) describes applications in group theory.
Read more about this topic: Greedy Algorithm For Egyptian Fractions
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