TY - JOUR

T1 - The Diophantine equation 2x2 + 1 = 3n

AU - Leu, Ming Guang

AU - Li, Guan Wei

PY - 2003/12

Y1 - 2003/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0344497406&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-03-07212-5

DO - 10.1090/S0002-9939-03-07212-5

M3 - 期刊論文

AN - SCOPUS:0344497406

VL - 131

SP - 3643

EP - 3645

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 12

ER -