Problems on Periodic Simple Continued Fractions

• Published in: Proceedings of the National Academy of Sciences - PNAS Vol. 69; no. 12; p. 3745
• Main Authors: Chowla, P., Chowla, S.
• Format: Journal Article
• Language: English
• Published: United States National Academy of Sciences of the United States of America 01.12.1972; National Acad Sciences
• Subjects: Continued fractions; Integers; Physical Sciences: Mathematics
• ISSN: 0027-8424; 1091-6490
• DOI: 10.1073/pnas.69.12.3745

Let N be a positive non-square integer and a1,a2,... ,asbe the partial denominators in the period of length s = s(N) of the continued fraction for$\sqrt{N}$. Also let ΣN= as- as-1+ - ... ± a1, and let h(d) be the class-number of Q($\sqrt{d}$). Hirzebruch (unpublished) recently found the surprising theorem (which is a special case of more general results): If p is a prime$\equiv $3(4) and p > 3, then h(p) = 1 implies that Σp= 3h(-p). This result led to related conjectures presented herein.