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The Diophantine equation \(x^4+2^ny^4=1\) in quadratic number fields. (English) Zbl 1472.11095

One of the classical Diophantine equations are the equations of type \(x^4+y^4=1\) and its generalizations in different directions. It was A. Aigner [Jahresber. Dtsch. Math.-Ver. 43, 226–228 (1934; JFM 60.0128.01)] who considered the quadratic integer silutions of this equation and showed that nontrivial quadratic solutions exist only in \(\mathbb Q(\sqrt{-7})\).
V. A. Lebesgue [J. Math. Pures Appl. 18, 73–86 (1853)] investigated the integer solutions of \(x^4\pm 2^ny^4=z^2\).
Using a method of L. J. Mordell [Acta Arith. 14, 347–355 (1968; Zbl 0191.04904)] the author characterizes the quadratic integer solutions of \(x^4+ 2^ny^4=1\).

MSC:

11D25 Cubic and quartic Diophantine equations
11R11 Quadratic extensions
11D45 Counting solutions of Diophantine equations
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References:

[1] Aigner, A., ‘Über die möglichkeit von \({x}^4+{y}^4={z}^4\) in quadratischen körpern’, Jahresber. Dtsch. Math.-Ver.43 (1934), 226-229. · Zbl 0008.29502
[2] Faddeev, D. K., ‘Group of divisor classes on the curve defined by the equation \({x}^4+{y}^4=1\)’, Soviet Math. Dokl.1 (1960), 1149-1151. · Zbl 0100.03401
[3] Lebesgue, V. A., ‘Résolution des équations biquadratiques \((1), (2) {z}^2={x}^4\pm{2}^m{y}^4 , (3) {z}^2={2}^m{x}^4-{y}^4 , (4), (5) {2}^m{z}^2={x}^4\pm{y}^4\)’, J. Math. Pures Appl.18 (1853), 73-86. https://archive.org/details/s1journaldemat18liou/page/72/mode/2up.
[4] Manley, E. D., ‘On quadratic solutions of \({x}^4+p{y}^4={z}^4\)’, Rocky Mountain J. Math.36(3) (2006), 1027-1031. · Zbl 1139.11018
[5] Mordell, L. J., ‘The Diophantine equation \({x}^4+{y}^4=1\) in algebraic number fields’, Acta Arith.14 (1967/68), 347-355. · Zbl 0191.04904
[6] , ‘The L-functions and modular forms database’, http://www.lmfdb.org, (online; accessed 28 July 2020).
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