×

On two questions of Ono. (English) Zbl 1036.11006

The authors provide an elementary proof of a well-known result: Theorem 1: If \((T,U)\) is the fundamental solution of \(x^2-py^2=1\), where \(p\) is prime, then \(T\equiv 1\pmod p\) if and only if \(p=2\) or \(p \equiv 7\pmod 8\). They also provide a variant of Lagrange’s theorem on the solvability of \(x^2-py^2=\pm 1\) using continued fractions.

MSC:

11D09 Quadratic and bilinear Diophantine equations
11A55 Continued fractions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Friesen, C.: Legendre symbols and continued fractions. Acta Arith., 59 (4), 365-379 (1991). · Zbl 0706.11004
[2] Halter-Koch, F.: Über Pellsche Gleichungen und Kettenbrüche. Arch. Math. (Basel), 49 (1), 29-37 (1987). · Zbl 0631.10008 · doi:10.1007/BF01200225
[3] Mollin, R. A.: Quadratics. CRC Press, Boca Raton-New York-London-Tokyo (1996).
[4] Ono, T.: On certain exact sequences for \(\Gamma_0(m)\). Proc. Japan Acad., 78A , 83-86 (2002). · Zbl 1106.11302 · doi:10.3792/pjaa.78.83
[5] Ono, T.: An email to H. Stark. August 10 (2002).
[6] Rose, H. E.: A Course in Number Theory. Oxford Univ. Press, Oxford (1988). · Zbl 0637.10002
[7] Williams, H. C.: Some results concerning the nearest integer continued fraction expansion of \(\sqrt D\). J. Reine Angew. Math., 315 , 1-15 (1980). · Zbl 0425.10037 · doi:10.1515/crll.1980.315.1
[8] Yamamoto, Y.: On the class number problem of quadratic fields. Sugaku, 40 , 167-174 (1988). (In Japanese). · Zbl 0664.12005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.