On the Diophantine equation \(x^2 - Dy^2 = n\). (English) Zbl 1281.11026
The author uses the method of his joint paper with F. Xu [Proc. Lond. Math. Soc. (3) 104, No. 5, 1019–1044 (2012; Zbl 1281.11065)] to obtain explicit conditions for the solvability of the title equation for \(n\in\mathbb Z\). In addition, they need to construct explicit abelian extensions of \(\mathbb Q(\sqrt D)\) corresponding to the idelic class groups. His main results cover the following two cases:
(1) \(D = pq\), where \(p, q \equiv 1 \bmod 4\) are distinct primes with \((\frac{q}{p}) = 1 \) and \((\frac{p}{q})_4 (\frac{q}{p})_4 = - 1\).
(2) \(D = 2p_1p_2\cdots p_m\), where \(p_i \equiv 1\bmod 8\), \(1\leq i\leq m\) are distinct primes and \(D = r^2+s^2\) with \(r, s \equiv \pm3 \bmod 8\).
(1) \(D = pq\), where \(p, q \equiv 1 \bmod 4\) are distinct primes with \((\frac{q}{p}) = 1 \) and \((\frac{p}{q})_4 (\frac{q}{p})_4 = - 1\).
(2) \(D = 2p_1p_2\cdots p_m\), where \(p_i \equiv 1\bmod 8\), \(1\leq i\leq m\) are distinct primes and \(D = r^2+s^2\) with \(r, s \equiv \pm3 \bmod 8\).
Reviewer: Olaf Ninnemann (Berlin)
Citations:
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