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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
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Abstract

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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

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