Problems on periodic simple continued fractions

Abstract

Let N be a positive non-square integer and a₁,a₂,…,a₃ be the partial denominators in the period of length s = s(N) of the continued fraction for √N. Also let ∑ N = a₃ - a_3-1 + −… ± a₁, and let h(d) be the class-number of Q(√d). Hirzebruch (unpublished) recently found the surprising theorem (which is a special case of more general results): If p is a prime≡ 3(4) and p > 3, then h(p) = 1 implies that ∑ p = 3h(−p).