Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 328.10010
Autor: Bleicher, Michael N.; Erdös, Paul
Title: Denominators of Egyptian fractions. (In English)
Source: J. Number Theory 8, 157-168 (1976).
Review: The authors obtain, by elementary methods, good upper and lower bounds for the size of the denominators of Egyptian expansions of fractions and also state several related conjectures. A fraction a/b is said to be written in Egyptian form if we write a/b = 1/n1+1/n2+...+1/nk, n1 < n2 < ... < nk, where the ni are positive integers. Let D(a,b) be the minimal value of nk in all expansions of a/b. Let D(b) be given by D(b) = max {D(a,b): 0 < a < b }. In this work it is shown that D(b) \leq Kb(ln b)3 for some constant K and that for P a prime D(P) \geq P {{ log2P } } where {{x } } = -[-x] is the least integer not less than x. Both theoretical and computational evidence are given to indicate that D(N)/N is maximum when N is a prime. A number of special cases are dealt with, for example, the authors prove that D(Pn) < 2Pn-1D(P). Among the conjectures stated the two of most general interest are, perhaps, (i) D(N) is submultiplicative, i.e., D(N · M) \leq D(N) · D(M). If true, relative primeness of M and N is probably irrelevant. (ii) Let n1 < n2 < ... be an infinite sequence of positive integers such that ni+1/ni > c > 1. Can the set of rationals a/b for which a/b = 1/ni1+1/ni2+...+1/nit is solvable for some t contain all the rationals in some interval (\alpha , \beta). We conjecture not. The main results have been improved upon in a second paper by the same authors [Illinois J. Math. 20, 598-613 (1976; Zbl 336.10007.]
Classif.: * 11A63 Radix representation
11D85 Representation problems of integers
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