Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 336.10007
Autor: Bleicher, Michael N.; Erdös, Paul
Title: Denominators of Egyptian fractions. II. (In English)
Source: Ill. J. Math. 20, 598-613 (1976).
Review: [Part I, cf. J. Number Theory 8, 157-168 (1976; Zbl 328.10010).] A positive fraction a/N is said to be written in Egyptian form if we write a/N = 1/n1+1/n2+...+1/nk, 0 < n1 < n2 < ... < nk, where the ni are integers. Among the many expansions for each fraction a/N there is some expansion for which nk is minimal. Let D(a,N) denote the minimal value of nk. Define D(N) by D(N) = max {D(a,N): 0 < a < N }. We are interested in the behavior of D(N). In this paper we show that on the one hand for a prime P large enough that log2rP \geq 1, D(P) \geq {P log P log2P \over logr+1P sum r+1j = 4 logjP} and on the other hand that for \epsilon > 0 and N sufficiently large (Theorem 1 and its corollary yield more precise statements),
D(N) \leq (1+\epsilon)N(log N)2. We conjecture that the exponent 2 can be replaced by (1+\delta) for \delta > 0. As part of the proof of the above results we need to analyze the number of distinct subsums of the series sum Ni = 11/i, say S(N). We show that whenever log2rN \geq 1,
{\alpha N \over log N} prod rj = 3 logjN \leq log S(N) \leq {N logrN \over log N} prod rj = 3 logjN for some \alpha \geq 1/e.
Classif.: * 11A63 Radix representation
11D85 Representation problems of integers
11D61 Exponential diophantine equations
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