The Diophantine equation 2x2 + 1 = 3n

Ming Guang Leu, Guan Wei Li

Research output: Contribution to journalArticlepeer-review

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Abstract

Let p be a rational prime and D a positive rational integer coprime with p. Denote by N (D,1, p) the number of solutions (x, n) of the equation Dx 2 + 1 = pn in rational integers x ≥ 1 and n ≥ 1. In a paper of Le, he claimed that N (D, 1, p) ≤ 2 without giving a proof. Furthermore, the statement N(D, 1, p) ≤ 2 has been used by Le, Bugeaud and Shorey in their papers to derive results on certain Diophantine equations. In this paper we point out that the statement N (D,1, p) ≤ 2 is incorrect by proving that N (2, 1, 3) = 3.

Original languageEnglish
Pages (from-to)3643-3645
Number of pages3
JournalProceedings of the American Mathematical Society
Volume131
Issue number12
DOIs
StatePublished - Dec 2003

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