Congruences modulo 4 of calibers of real quadratic fields (Q2888162)

The authors establish some results about the \(m\)-calibers of quadratic irrationalities of discriminants \(8p\) and \(pq\), where \(p\) and \(q\) are prime numbers congruent to \(1\bmod 4\), and \(p\nmid q\). The \(m\)-caliber of a discriminant \(D\) of a quadratic irrationality, denoted by \(k^+(D)\), can be described as the cardinality of the set \(\text{Cl}^*(D)\) of all quadratic irrationalities \(W\) of discriminant \(D\) such that the ``minus'' continued fraction expansion of \(W\) is purely periodic. It is well-known that the cardinality of the set of \(\mathrm{SL}_2(\mathbb Z)\)-equivalence classes of \(\text{CL}^+(D)\) is the narrow class number of discriminant \(D\), and the authors compare their \(m\)-caliber results to corresponding narrow class number results in regard to divisibility of 4.NEWLINENEWLINE NEWLINETo establish their results, they use various properties of continued fractions and related matrices, a detailed analysis of parity relations, and make the assumption that the norms of the fundamental units of their fields \(\mathbb Q(\sqrt{2p})\) and \(\text{Cl}(\sqrt{p,q})\) are both \(-1\). Letting \(p= X_p^2+ Y_p^2\) and \(q= X_q^2+ Y_q^2\) with \(OLX_pLY_p\) and \(OLX_q LY_q\), their first result is stated as \(k^+(8p)\equiv 1-(-1)^{X_p}\pmod 4\).NEWLINENEWLINE NEWLINEFor their second result the authors make the additional assumption that \(X_p\not\equiv X_q\pmod 2\), and they utilize the Kronecker symbol \(({q\over p})\) and the lengths \(\ell(a)\) and \(\ell(b)\) of the periods of the purely periodic palindromic continued fraction expansions of the two quadratic numbers \(a\) and \(b\), that correspond to the two decompositions of \(pq\) into sums of squares, where \(a\) and \(b\) are quadratic irrationalities of discriminant \(pq\) whose continued fraction expansions are purely periodic. The authors state their second result as follows: NEWLINE\[NEWLINEk^+ (pq)\equiv 1+\text{sgn}(X_p Y_q- Y_p X_q)(-1)\Biggl[X_p+ {(\ell(a)- \ell(b))\over 2}\Biggr] \Biggl({q\over p}\Biggr)\pmod 4.NEWLINE\]NEWLINE NEWLINEWe note that to prove their result, the authors provide a detailed analysis for some particular cases, and then leave the details of the remaining cases to the reader. We also note that there are two omissions of prime superscript symbols in the proofs of the results, regarding the representations of equivalence classes of the sets of quadratic irrationalities of \(8p\) and \(pq\) whose continued fraction expansions are purely periodic. However, these notational errors have no effect on the proofs of the results, which are correct for the cases that are illustrated.