The continued fractions of the square roots of integers: on the parity of the length of their period (Q611959)

The author studies negative Pell equation \(x^2-Qy^2=-1\) using continued fraction expansion of \(\sqrt{Q}\). Some results from this article are known. First theorems belong to Legendre (see the history of negative Pell equation in [\textit{L. E. Dickson}, History of the theory of numbers. Vol. II. Diophantine analysis. Washington: Carnegie Institution of Washington (1920; JFM 47.0100.04) p. 365; reprint Dover (2005; Zbl 1214.11002)], and [\textit{I. Kh. I. Gerasim}, Istor.-Mat. Issled. 32/33, 199--211 (1990; Zbl 0731.01014)] for additional bibliography). Lemmas D and E are very similar to Galois theorem on purely periodic continued fraction [\textit{É. Galois}, Ann. Math. Pures Appl. 19, 294--301 (1828/29), \url{https://www.bibnum.education.fr/sites/default/files/galois-texte.pdf}]. The last theorem is seems to be new: Let \(\Sigma\) be the sum of the elements in the period of \(\sqrt{Q}\). If \(\Sigma\) is even, then either equation \(x^2-Qy^2=-1\) is solvable or \(Q\) is not prime.