On the cyclotomic polynomial
This paper is divided into two parts. In the first part some of the author's previous results [J. Number Theory 16, 420--429 (1983; Zbl 0515.12004)] are generalized. E.g., if \(2<m\in\mathbb Z\), \(1 <n\in\mathbb Z\), \(\Phi_m\) the \(m\)-th cyclotomic polynomial, \(\Phi_m(x)\equiv 0\pmod n\) for some \(x\in\mathbb Z\), then there are integers \(a, b\) with \(n=a^2-m^*b^2\). This is proven under some restrictions on \(n\), and the right definition of \(m^*\) in relation to \(m\), when the class number of \(\mathbb Q(\sqrt{m^*})\) equals one. There are counter-examples that these restrictions are necessary. As application this theorem gives some well-known results of Fermat and the following: If \(p\) is an odd prime then \(x^2 - py^2\) represents all primes \(\equiv -1\pmod{4p}\). It is well-known that the only coefficients of \(\Phi_{pq}(X)\) \((p, q\) distinct primes) are \(0, 1, -1\). In the second part the author determines explicitly the right sign for each power of \(X\).
representation of integers by binary quadratic forms
scientific article; zbMATH DE number 3807691