Problems on Periodic Simple Continued Fractions

Abstract

Let N be a positive non-square integer and a(1),a(2),...,a(3) be the partial denominators in the period of length s = s(N) of the continued fraction for radicalN. Also let Sigma(N) = a(3) - a(3-1) + -... +/- a(1), and let h(d) be the class-number of Q( radicald). Hirzebruch (unpublished) recently found the surprising theorem (which is a special case of more general results): If p is a prime identical with3(4) and p > 3, then h(p) = 1 implies that Sigma(p) = 3h(-p).This result led to related conjectures presented herein.