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An exploration of hyperpower equations nx = ny

Published online by Cambridge University Press:  01 August 2016

Roger Voles
Roger Voles*
Affiliation:
49 Park Road, Chiswick, London W4 3EYrogervoles@mcmail.com
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Extract

My interest in this fascinating area of mathematics was aroused by a recent article [1] in which it was shown that

has an infinity of non-trivial solutions (i.e. u ≠ v) in positive integers, rationals and algebraic irrationals. Replacing u and v by their reciprocals, x and y, permits the equation to be written as

Type
Articles
Information
The Mathematical Gazette , Volume 83 , Issue 497 , July 1999 , pp. 210 - 215
Copyright
Copyright © The Mathematical Association 1999

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References

1. Churchhouse, R. F., Solutions of the equation xy = Yx , Bulletin of the IMA, 31 (July/August 1995) p. 106.Google Scholar
2. Knoebel, R. A., Exponentials reiterated, Amer. Math. Monthly 88 (1981) pp. 235–252.Google Scholar
3. Euler, L. (edited by Faber, Georg), De formulis exponentialibus replicatis, Euleri Opera Omnia, 2 (1927) pp. 268–297.Google Scholar
5. Ash, J. M., The Limit of [nx] as x tends to zero, Mathematics Magazine, 69 (June 1996) pp. 207–209.Google Scholar