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On recursive solutions of a unit fraction equation

Part of: Sequences and sets Multiplicative number theory Computational number theory Diophantine equations

Published online by Cambridge University Press:  09 April 2009

Lawrence Brenton  and
Robert R. Bruner
Lawrence Brenton
Affiliation:
Wayne State University, Detroit, Michigan 48202, U.S.A., brenton@math.wayne.edu
Robert R. Bruner
Affiliation:
Wayne State University, Detroit, Michigan 48202, U.S.A., rrb@math.wayne.edu
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Abstract

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We consider the Egyptian fraction equation and discuss techniques for generating solutions. By examining a quadratic recurrence relation modulo a family of primes we have found some 500 new infinite sequences of solutions. We also initiate an investigation of the randomness of the distribution of solutions, and show that there are infinitely many solutions not generated by the aforementioned technique.

MSC classification

Secondary: 11D68: Rational numbers as sums of fractions 11N45: Asymptotic results on counting functions for algebraic and topological structures 11B50: Sequences (mod $m$) 11Y50: Computer solution of Diophantine equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 57 , Issue 3 , December 1994 , pp. 341 - 356
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Barbeau, E., ‘Expressing one as a sum of distinct reciprocals: comments and a bibliography’, Eureka(Crux Math.) 3 (1977), 178–181.Google Scholar
[2]Brenton, L. and Hill, R., ‘On the diophantine equation and a class of homologically trivial complex surface singularities’, Pacific J. Math. 133 (1988), 41–67.Google Scholar
[3]Burshtein, N., ‘On distinct unit fractions whose sum equals 1’, Discrete Math. 5 (1973), 201–206.Google Scholar
[4]Cao, Z., Liu, R. and Zhang, L., ‘On the equation and Znám's problem’, J. Number Theory 27 (1987), 206–211.Google Scholar
[5]Curtiss, D., ‘On Kellogg's diophantine problem’, Amer. Math. Monthly 29 (1922), 380–387.CrossRefGoogle Scholar
[6]Grauert, H., ‘Über modifikationen und exzeptionelle analytische Mengen’, Math. Ann. 146 (1962), 331–368.Google Scholar
[7]Johannessen, D. M. and Søhus, T. V., ‘On stambrøker’, Nord. Mat. Tidskr. 22 (1974), 103–107.Google Scholar
[8]Knuth, D., The art of computer programming, volume 2 (Addison-Wesley, Reading, 1968).Google Scholar
[9]Sun, Q., ‘On a problem of Znám’, Sichuan Daxue Xuebao 4 (1983), 9–11.Google Scholar
[10]Sylvester, J., ‘On a point in the theory of vulgar fractions’, Amer. J. Math. 3 (1880), 332–335, 388–389.Google Scholar