Glasnik Matematicki, Vol. 47, No. 1 (2012), 53-59.

A NOTE ON THE SIMULTANEOUS PELL EQUATIONS X2-AY2=1 AND Z2-BY2=1

Maohua Le

Department of Mathematics, Zhanjiang Normal College, Zhanjiang, Guangdong 524048, P.R. China
e-mail: lemaohua2008@163.com

Abstract.   Let m,n be positive integers with 1 < m < n. Let δ be a positive number with 1/2 < δ < 1 . In this paper we prove that if gcd(m,n)>nδ and n>(8× 1016(log(1016/θ3))3/θ3)1/θ, where θ=min(1-δ, 2δ-1), then the simultaneous Pell equations x2-(m2-1)y2=1 and z2-(n2-1)y2=1 have only one positive integer solution (x,y,z)=(m,1,n).

2010 Mathematics Subject Classification.   11D09.

Key words and phrases.   Simultaneous Pell equations; number of solutions.

Full text (PDF) (free access)

DOI: 10.3336/gm.47.1.04

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