Residue properties of certain quadratic units
The author proves the following two theorems, thereby establishing conjectures of \textit{P. A. Leonard} and the reviewer [Pac. J. Math. 71, 101-106 (1977; Zbl 0335.12006), Rocky Mt. J. Math. 9, 683-692 (1979; Zbl 0386.12002) and of \textit{E. Lehmer} [J. Reine Angew. Math. 268/269, 294-301 (1974; Zbl 0289.12007)]. - Theorem 1. Let p and q be primes such that \(p\equiv 1 (mod 8)\), \(q\equiv 7 (mod 8)\), and \((q/p)=1\). The class number of \(K={\mathbb{Q}}(\sqrt{-q})\) is odd and \(p^ k\) has the form \(a^ 2+16qb^ 2\). The class number of \(L={\mathbb{Q}}(\sqrt{-2q})\) is \(4\ell\), where \(\ell\) is odd if and only if \(q\equiv 7 (mod 16)\). Let \(\epsilon\) be the fundamental unit of \({\mathbb{Q}}(\sqrt{2})\). Then \(p^{\ell}\) is represented by one of the following forms, and the corresponding character formula holds: \[ p^{\ell}=x^ 2+8qy^ 2\quad and\quad (\epsilon /p)=(-1)^ b(2/p)_ 4;\quad p^{\ell}=2x^ 2+qy^ 2\quad and\quad (\epsilon /p)=(-1)^{b+1}(2/p)_ 4. \] Theorem 2. Let p and q be primes such that \(p\equiv 1 (mod 4)\), \(q\equiv 5 (mod 8)\), and \((q/p)=1\). Let \(\epsilon\) be the fundamental unit of \({\mathbb{Q}}(\sqrt{q})\). The class number of \(K={\mathbb{Q}}(\sqrt{-q})\) is 2k, where k is odd, and \(p^ k\) has the form \(x^ 2+qy^ 2\). Furthermore \((\epsilon /p)=(- 1)^ y\). Suppose that in fact \((\epsilon /p)=1\) and that \(p\equiv 1 (mod 8)\). Then \(y=4d\). Write p in the form \(a^ 2+16b^ 2\). Then \((\epsilon /p)_ 4=(-1)^{b+d}\).
representation of powers of primes by quadratic forms
scientific article; zbMATH DE number 3904700
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