Uniqueness conjecture on simultaneous Pell equations
Volume 207 / 2023
Acta Arithmetica 207 (2023), 279-296
MSC: Primary 11D09; Secondary 11B37, 11J68, 11J86.
DOI: 10.4064/aa221005-1-3
Published online: 11 April 2023
Abstract
Let $A$ and $B$ be distinct positive integers. It is known that any positive solution of the simultaneous Pell equations $x^2-Ay^2=1$ and $z^2-By^2=1$ gives rise to a positive solution of the simultaneous Pell equations $x^2-(m^2-1)y^2=1$ and $z^2-(n^2-1)y^2=1$ for some distinct integers $m$ and $n$ greater than 1. In this paper, we prove that the latter equations have only the positive solution $(x,y,z)=(m,1,n)$ if there exist positive integers $a$, $b$, $c$ with $a \lt b$ satisfying $ac+1=m^2$, $bc+1=n^2$ and $c \ge 5b^4$. Moreover, we show that the same conclusion holds if we replace the assumption $c \ge 5b^4$ above by $a=1$ and $b=b_0^2-1=p^{e}+1$ with $b_0,\,e$ positive integers and $p$ a prime.
