On the diophantine equation $x^y - y^x = c^z$

Zhongfeng Zhang; Jiagui Luo; Pingzhi Yuan

On the diophantine equation

x^{y} - y^{x} = c^{z}

Zhongfeng Zhang ; Jiagui Luo ; Pingzhi Yuan

Colloquium Mathematicae, Tome 126 (2012), p. 277-285 / Harvested from The Polish Digital Mathematics Library

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Résumé

Applying results on linear forms in p-adic logarithms, we prove that if (x,y,z) is a positive integer solution to the equation $x^{y} - y^{x} = c^{z}$ with gcd(x,y) = 1 then (x,y,z) = (2,1,k), (3,2,k), k ≥ 1 if c = 1, and either $(x, y, z) = (c^{k} + 1, 1, k)$ , k ≥ 1 or $2 \leq x < y \leq m a x 1.5 \times 10^{10}, c$ if c ≥ 2.

Publié le : 2012-01-01

Zbl 1297.11017

EUDML-ID : urn:eudml:doc:283734

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Zhongfeng Zhang; Jiagui Luo; Pingzhi Yuan. On the diophantine equation $x^y - y^x = c^z$
            . Colloquium Mathematicae, Tome 126 (2012) pp. 277-285. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm128-2-13/