Mathematics of Computation

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ISSN 1088-6842(e) ISSN 0025-5718(p)

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

Author(s): 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:

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 , Vid-Akad. Skr. Norske Oslo 1942 No. 9.

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