Functiones et Approximatio Commentarii Mathematici

On the diophantine equation $2^x=x^2+y^2-2$

Alexandru Gica and Florian Luca

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In this paper, we show that the only positive integer solutions of the equation $2^x=x^2+y^2-2$ are $(x,y)=(3,1),~(5,3),~(7,9)$. We propose also the following conjecture: the equation $2^x=y^2+z^2(x^2-2)$, where $y,z$ are odd positive integers and $x$ is a positive integer such that $x^2-2$ is a prime number, has the only solutions $(x,y,z)=(3,1,1),~(5,3,1),~(7,9,1),~(13,3,7)$. The conjecture implies a recent result of Lee [4] which states that if $x^2-2$ is an odd prime number such that the class number $h(x^2-2)$ of the quadratic field $\mathbb{Q}[{\sqrt{x^2-2}}]$ is $1$, then $x=3,5,7,13$.

Article information

Funct. Approx. Comment. Math., Volume 46, Number 1 (2012), 109-116.

First available in Project Euclid: 30 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D61: Exponential equations
Secondary: 11J70: Continued fractions and generalizations [See also 11A55, 11K50] 11J86: Linear forms in logarithms; Baker's method

diophantine equations applications of Baker's method


Gica, Alexandru; Luca, Florian. On the diophantine equation $2^x=x^2+y^2-2$. Funct. Approx. Comment. Math. 46 (2012), no. 1, 109--116. doi:10.7169/facm/2012.46.1.8.

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