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On the fourth-powerfree part of x2 + 2

Published online by Cambridge University Press:  18 May 2009

Benjamin M. M. de Weger
Benjamin M. M. de Weger
Affiliation:
Sportsingel 30, 2924 XN Krimpen aan den Ijssel, The Netherlands, E-mail: dweger@xs4all.nl
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Abstract

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We show that x = 59 is the largest positive integer for which the fourth-powerfree part of x2 + 2 is at most 100. This implies the solution of the problem, posed recently by J. H. E. Cohn, to prove that (x, y) = (1, 1) is the only solution in nonnegative integers to the diophantine equation x2 – 3y4 = –2, as well as a new solution to the problem, posed a long time ago by the same J. H. E. Cohn and solved before by R. Bumby and N. Tzanakis, to prove that (x, y) = (1, 1), (11, 3) are the only solutions in nonnegative integers to the diophantine equation 2x2 – 3y4 = – 1.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 40 , Issue 3 , September 1998 , pp. 299 - 310
Copyright
Copyright © Glasgow Mathematical Journal Trust 1998

References

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