Functiones et Approximatio Commentarii Mathematici

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

Alexandru Gica and Florian Luca

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 30 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.facm/1333112937

Digital Object Identifier
doi:10.7169/facm/2012.46.1.8

Mathematical Reviews number (MathSciNet)
MR2951732

Zentralblatt MATH identifier
1283.11061

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

Keywords
diophantine equations applications of Baker's method

Citation

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. https://projecteuclid.org/euclid.facm/1333112937


Export citation

References

  • Y. Bugeaud and M. Laurent, Minoration effective de la distance $p$-adique entre puissances de nombres algébriques, J. of Number Theory 61 (1996), 311–342.
  • A. Dujella, Continued fractions and RSA with small secret exponent, Tatra Mt. Math. Publ. 29 (2004), 101–112.
  • A. Gica, An additive problem, An. Univ. Buc. Mat. 53 (2004), 229–234.
  • J. Lee, The complete determination of wide Richaud-Degert types which are not $5$ modulo $8$ with class number one, Acta Arith. 140 (2009), 1–29.
  • R. A. Mollin, Quadratics, CRC Press, 1996.
  • R. T. Worley, Estimating $|\alpha-p/q|$, J. Austral. Math. Soc. 31 (1981), 202–206.
  • D. Zagier, Nombres de classes et fractions continues, Astérisque 24-25 (1975), 81–97.