Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T15:32:26.217Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE DIOPHANTINE EQUATION z2=f(x)2±f(y)2, II

Part of: Diophantine equations

Published online by Cambridge University Press:  22 June 2010

BO HE ,
ALAIN TOGBÉ  and
MACIEJ ULAS
BO HE
Affiliation:
Department of Mathematics, ABa Teacher’s College, Wenchuan, Sichuan 623000, PR China (email: bhe@live.cn)
ALAIN TOGBÉ*
Affiliation:
Mathematics Department, Purdue University North Central, 1401 S, U.S. 421, Westville IN 46391, USA (email: atogbe@pnc.edu)
MACIEJ ULAS
Affiliation:
Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland (email: Maciej.Ulas@im.uj.edu.pl)
*
✉For correspondence; e-mail: atogbe@pnc.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let f∈ℚ[X] and let us consider a Diophantine equation z2=f(x)2±f(y)2. In this paper, we continue the study of the existence of integer solutions of the equation, when the degree of f is 2 and if f(x) is a triangular number or a tetrahedral number.

Keywords

Pythagorean triplesDiophantine equations

MSC classification

Secondary: 11D25: Cubic and quartic equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 2 , October 2010 , pp. 187 - 204
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The first author was supported by the Applied Basic Research Foundation of Sichuan Provincial Science and Technology Department (No. 2009JY0091). The second author thanks Purdue University North Central for the support. The third author was supported by START for young scientists, founded by the Polish Science Foundation.

References

[1]Dujella, A., ‘Continued fractions and RSA with small secret exponent’, Tatra Mt. Math. Publ. 29 (2004), 101–112.Google Scholar
[2]Dujella, A. and Jadrijević, B., ‘A family of quartic Thue inequations’, Acta Arith. 111 (2004), 61–76.CrossRefGoogle Scholar
[3]Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 4th edn (Clarendon Press, Oxford, 1960).Google Scholar
[4]Sierpiński, W., Triangular Numbers, Biblioteczka Matematyczna, 12 (PZWS, Warsaw, 1962) (in Polish).Google Scholar
[5]Ulas, M., ‘On the Diophantine equation f(x)f(y)=f(z)2’, Colloq. Math. 107 (2007), 1–6.CrossRefGoogle Scholar
[6]Ulas, M. and Togbé, A., ‘On the Diophantine equation z 2=f(x)2±f(y)2’, Publ. Math. Debrecen 76(1–2) (2010), 183–201.CrossRefGoogle Scholar
[7]Worley, R. T., ‘Estimating ∣α−p/q∣’, J. Aust. Math. Soc. 31 (1981), 202–206.CrossRefGoogle Scholar