The Diophantine equation $x^4+1=Dy^2$
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- by J. H. E. Cohn PDF
- Math. Comp. 66 (1997), 1347-1351 Request permission
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
- J. H. E. Cohn, Lucas and Fibonacci numbers and some Diophantine equations, Proc. Glasgow Math. Assoc. 7 (1965), 24–28 (1965). MR 177944, DOI 10.1017/S2040618500035115
- 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.
Additional Information
- J. H. E. Cohn
- Affiliation: Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom
- Email: j.cohn@rhbnc.ac.uk
- Received by editor(s): March 4, 1996
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp. 66 (1997), 1347-1351
- MSC (1991): Primary 11D25
- DOI: https://doi.org/10.1090/S0025-5718-97-00851-X
- MathSciNet review: 1415800