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On the equation \(y^x\pm x^y=z^2\). (English) Zbl 1014.11024

The authors prove that the equation \(y^x\pm x^y= z^2\) has only the positive integer solution \((x,y,z)= (2,3,1)\) satisfying \(\min(x,y)> 1\), \(\gcd(x,y)= 1\) and \(2\mid xy\). The proofs depend on some lower bounds of linear forms in \(p\)-adic and Archimedean logarithms.

MSC:

11D61 Exponential Diophantine equations
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References:

[1] Y. Bugeaud and M. Laurent, Minoration effective de la distance \(p\)-adique entre puissances de nombres algébriques , J. Number Theory 61 (1996), 311-342. · Zbl 0870.11045 · doi:10.1006/jnth.1996.0152
[2] M. Laurent, M. Mignotte and Y. Nesterenko, Formes linéaires en deux logarithmes et déterminants d’interpolation , J. Number Theory 55 (1995), 285-321. · Zbl 0843.11036 · doi:10.1006/jnth.1995.1141
[3] M. Mignotte, A corollary to a theorem of Laurent-Mignotte-Nesterenko , Acta Arit. 86 (1998), 101-111. · Zbl 0919.11051
[4] P. Ribenboim, Catalan’s conjecture , Academic Press, Boston, 1994. · Zbl 0824.11010
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