On the Diophantine equation 𝑥²ⁿ-𝒟𝓎²=1

Cao, Zhen Fu

doi:10.1090/S0002-9939-1986-0848864-4

On the Diophantine equation $x^ {2n}-\mathcal {D}y^2=1$
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by Zhen Fu Cao

Proc. Amer. Math. Soc. 98 (1986), 11-16

DOI: https://doi.org/10.1090/S0002-9939-1986-0848864-4

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Abstract:

In this paper, it has been proved that if $n > 2$ and Pell’s equation ${u^2} - \mathcal {D}{v^2} = - 1$ has integer solution, then the equation ${x^{2n}} - \mathcal {D}{y^2} = 1$ has only solution in positive integers $x = 3$, $y = 22$ (when $n = 5$, $\mathcal {D} = 122$). That is proved by studying the equations ${x^p} + 1 = 2{y^2}$ and ${x^p} - 1 = 2{y^2}$ ($p$ is an odd prime). In addition, some applications of the above result are given.

References

Chao Ko and Chi Sun, On the Diophantine equation $x^{4}-Dy^{2}=1$. II, Chinese Ann. Math. 1 (1980), no. 1, 83–89 (Chinese, with English summary). MR 591236
Zhen Fu Cao, On the Diophantine equations $x^{2}+1=2y^{2},$ $x^{2}-1=2Dz^{2}$, J. Math. (Wuhan) 3 (1983), no. 3, 227–235 (Chinese, with English summary). MR 747900
L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355

On the diophantine equation

1

Wilhelm Ljunggren, Some theorems on indeterminate equations of the form $x^n-1/x-1=y^q$, Norsk Mat. Tidsskr. 25 (1943), 17–20 (Norwegian). MR 18674
Robert W. van der Waall, On the Diophantine equations $x^{2}+x+1=3v^{2}$, $x^{3}-1=2y^{2}$, $x^{3}+1=2y^{2}$, Simon Stevin 46 (1972/73), 39–51. MR 316374
P. Erdös, On a Diophantine equation, J. London Math. Soc. 26 (1951), 176–178. MR 41156, DOI 10.1112/jlms/s1-26.3.176
W. Ljunggren, On the diophantine equation $Cx^{2}+D=2y^{n}$, Math. Scand. 18 (1966), 69–86. MR 204358, DOI 10.7146/math.scand.a-10781

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Proceedings of the American Mathematical Society

On the Diophantine equation $x^ {2n}-\mathcal {D}y^2=1$
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Abstract: