On the number of solutions of 𝑥²-4𝑚(𝑚+1)𝑦²=𝑦²-𝑏𝑧²=1

Yuan, Pingzhi

doi:10.1090/S0002-9939-04-07418-0

On the number of solutions of $x^2-4m(m+1)y^2=y^2-bz^2=1$
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Proc. Amer. Math. Soc. 132 (2004), 1561-1566 Request permission

Abstract:

In this paper, using a result of Ljunggren and some results on primitive prime factors of Lucas sequences of the first kind, we prove the following results by an elementary argument: if $m$ and $b$ are positive integers, then the simultaneous Pell equations \[ x^2-4m(m+1)y^2=y^2-bz^2=1\] possesses at most one solution $(x,y,z)$ in positive integers.

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