On simultaneous Pell equations \(x^2-(m^2-1) y^2=z^2-(n^2-1) y^2=1\). (Chinese. English summary) Zbl 07828134
Summary: Let \(m\), \(n\), \(L\) be positive integer. The following conclusion are proved: If \(m<n \leq m+Lm^\varepsilon\), \(\varepsilon \in (0,1)\), and \(m>(123789L \sqrt{L})^{\frac{1}{1-\varepsilon}}\), or \(j>10.25 \times 10^{12} \log^4 (2(L+1) (123789L (\sqrt{L})^{\frac{1}{1-\varepsilon}}\), then positive integer solutions of simultaneous Pell equations \(x^2 -(m^2 - 1) y^2 = z^2 -(n^2 - 1) y^2 = 1\) satisfy \(1 \leq k \leq \delta L^2\), where \(\delta \in [\frac{1}{2}(123787 L \sqrt{L})^{\frac{1}{ \varepsilon -1}}, 1]\), and
\[
y = \frac{(m+\sqrt{m^2-1})^j - (m-\sqrt{m^2-1})^j}{2 \sqrt{m^2 -1}} = \frac{(n+\sqrt{n^2-1})^k - (n-\sqrt{n^2-1})^k}{2 \sqrt{n^2 -1}},
\]
and \(j=k=1\) or \(k+2 \leq j < \frac{1}{3} (5 - 2 \varepsilon) k\), \(2 | (j+k)\), \(k > \frac{3}{1-\varepsilon}\). It improves the previous work of B. He et al. [Proc. Am. Math. Soc. 143, No. 11, 4685–4693 (2015; Zbl 1378.11041)].
MSC:
| 11D09 | Quadratic and bilinear Diophantine equations |
| 11J86 | Linear forms in logarithms; Baker’s method |
Citations:
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