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The convexity of the function y = E(x) defined by xy = yx

Published online by Cambridge University Press:  02 March 2020

Alan F. Beardon
Affiliation:
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB
Russell A. Gordon
Affiliation:
Department of Mathematics and Statistics, Whitman College, 345 Boyer Avenue, Walla Walla, WA 99362, USA e-mails: afb@dpmms.cam.ac.uk; gordon@whitman.edu

Extract

The set of solutions to the equation xy = yx has been studied extensively over the past three centuries, including work by well known mathematicians such as Daniel Bernoulli (1700–1782), Leonhard Euler (1707–1783), and Christian Goldbach (1690–1764). Various mathematicians have focused on the integer, rational, real, and complex solutions. For example, it has been shown (see [1]) that the equality 24 = 42 gives the only distinct integer solutions. Our exposition below presents some of the key ideas behind the positive real solutions to this equation and illustrates how rational solutions can be found. To learn more about the various solutions, the reader can consult the articles listed at the end of this paper, as well as the extensive references given in these articles. It is also possible to find some of this material on the Web.

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Articles
Copyright
© Mathematical Association 2020

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References

Gerrish, F., a b = b a: the positive integer solution, Math. Gaz. 76 (November 1992) p. 403.CrossRefGoogle Scholar
Cundy, H. M., x y = y x: an investigation, Math. Gaz. 71 (June 1987) pp. 131135.CrossRefGoogle Scholar
Siu, M., An interesting exponential equation, Math. Gaz. 60 (October 1976) pp. 213215.CrossRefGoogle Scholar
Sved, M., On the rational solutions of x y = y x, Math. Mag. 63 (1990) pp. 3033.CrossRefGoogle Scholar
Beardon, A., The real solutions of x y = y x, Math. Mag. 39 (1966) pp. 108111.Google Scholar
Lóczi, L., Two centuries of the equations of commutativity and associativity of exponentiation, Teaching Mathematics and Computer Science, 1-2 (2003) pp. 219233.CrossRefGoogle Scholar
Bennet, A. and Reznick, B., Positive rational solutions to x y = y mx; a number-theoretic excursion, Amer. Math. Monthly 111 (2004) pp. 1321.Google Scholar
Knoebel, R., Exponentials reiterated, Amer. Math. Monthly 88 (1981) pp. 235252.CrossRefGoogle Scholar

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