ON the diophantine equation x2n-Dy2 = 1

Cao Zhenfu

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

ON the diophantine equation x²ⁿ-D^y2 = 1

Cao Zhenfu^*

^*Corresponding author for this work

Harbin Institute of Technology

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

In this paper, it has been proved that if n > 2 and Pell's equation u² - D^v2 = -1 has integer solution, then the equation x²ⁿ - D^y2 = 1 has only solution in positive integers x = 3, y = 22 (when n = 5, D = 122). That is proved by studying the equations xp + 1 = 2^y2 and x^p - 1 = 2^y2 (p is an odd prime). In addition, some applications of the above result are given.

Original language	English
Pages (from-to)	11-16
Number of pages	6
Journal	Proceedings of the American Mathematical Society
Volume	98
Issue number	1
DOIs	https://doi.org/10.1090/S0002-9939-1986-0848864-4
State	Published - Sep 1986
Externally published	Yes

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title = "ON the diophantine equation x2n-Dy2 = 1",

abstract = "In this paper, it has been proved that if n > 2 and Pell's equation u2 - Dv2 = -1 has integer solution, then the equation x2n - Dy2 = 1 has only solution in positive integers x = 3, y = 22 (when n = 5, D = 122). That is proved by studying the equations xp + 1 = 2y2 and xp - 1 = 2y2 (p is an odd prime). In addition, some applications of the above result are given.",

author = "Cao Zhenfu",

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AU - Zhenfu, Cao

PY - 1986/9

Y1 - 1986/9

N2 - In this paper, it has been proved that if n > 2 and Pell's equation u2 - Dv2 = -1 has integer solution, then the equation x2n - Dy2 = 1 has only solution in positive integers x = 3, y = 22 (when n = 5, D = 122). That is proved by studying the equations xp + 1 = 2y2 and xp - 1 = 2y2 (p is an odd prime). In addition, some applications of the above result are given.

AB - In this paper, it has been proved that if n > 2 and Pell's equation u2 - Dv2 = -1 has integer solution, then the equation x2n - Dy2 = 1 has only solution in positive integers x = 3, y = 22 (when n = 5, D = 122). That is proved by studying the equations xp + 1 = 2y2 and xp - 1 = 2y2 (p is an odd prime). In addition, some applications of the above result are given.

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ER -

ON the diophantine equation x²ⁿ-D^y2 = 1

Abstract

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