Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T02:52:44.883Z Has data issue: false hasContentIssue false
  • English
  • Français

On a diophantine equation

Published online by Cambridge University Press:  17 April 2009

Fadwa S. Abu Muriefah  and
S. Akhtar Arif
Fadwa S. Abu Muriefah
Affiliation:
Department of MathematicsGirls College of EducationRiyadhSaudi Arabia
S. Akhtar Arif
Affiliation:
Department of MathematicsGirls College of EducationRiyadhSaudi Arabia
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.

In this paper the equation x2 + 32k = yn where n ≥ 3 is studied. For n = 3, it is proved that it has a solution only if k = 3K + 2 and then there is a unique solution x = 46 × 33K and y = 13 × 32K. For n > 3 theorems are proved which determine a large number of values of k and n for which this equation has no solution. It is proved that if this equation has a solution for n > 3, then n is odd and k = 2δ.k′ where δ ≥ 1, (2, δ) = 1, k′ ≡ 15 (mod 20) and all the primes divisors p of n are congruent to 11 (mod 12).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 57 , Issue 2 , April 1998 , pp. 189 - 198
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Arif, S.A. and Muriefah, F.S. Abu, ‘The diophantine equation x 2 + 2k = yn’, Internat. J. Math. Math. Sci. 20 (1997), 299–304.CrossRefGoogle Scholar
[2]Cohn, J.H., ‘The diophantine equation x 2 + 2k = yn’, Arch. Math. 59 (1992), 341–344.CrossRefGoogle Scholar
[3]Cohn, J.H., ‘The diophantine equation x 2 + C = yn’, Acta Arith. 65 (1993), 367–381.CrossRefGoogle Scholar
[4]Lebesgue, V.A., ‘Sur I'impossibilitié en nombres entiers de l'equation xm = y 2 + 1’, Von. Ann. Des. Math 9 (1850), 178–181.Google Scholar
[5]Ljunggren, W., ‘On the diophantine equation x 2 + p 2 = yn’, Kong. Norsk. Vid. Selskab Forh. Trond. 16 (1943), 27–30.Google Scholar
[6]Nagell, T., ‘On the diophantine equation x 2 + 8D = yn’, Ark. Math. 3 (1955), 103–112.CrossRefGoogle Scholar
[7]Störmer, , ‘L'equation m arctan 1/x +n arctan 1/y - k.π/4’, Bull. Soc. Math. France 27 (1899), 160–170.Google Scholar