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Mathematics of Computation

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The Diophantine equation $x^4+1=Dy^2$

Author: J. H. E. Cohn
Journal: Math. Comp. 66 (1997), 1347-1351
MSC (1991): Primary 11D25
MathSciNet review: 1415800
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Abstract: An effective method is derived for solving the equation of the title in positive integers $x$ and $y$ for given $D$ completely, and is carried out for all $D<100000$. If $D$ is of the form $m^4+1$, then there is the solution $x=m$, $y=1$; in the above range, except for $D=70258$ with solution $x=261$, $y=257$, there are no other solutions.

References [Enhancements On Off] (What's this?)

  • 1. J. H. E. Cohn, Lucas and Fibonacci numbers and some Diophantine equations, Proc. Glasgow Math. Assoc. 7 (1965), 24-28. MR 31:2202
  • 2. Wilhelm Ljunggren, Einige Sätze über unbestimmte Gleichungen von der Form $Ax^4+Bx^2+C=Dy^2$, Vid-Akad. Skr. Norske Oslo 1942 No. 9.

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Additional Information

J. H. E. Cohn
Affiliation: Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom

Received by editor(s): March 4, 1996
Article copyright: © Copyright 1997 American Mathematical Society

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