Problems on periodic simple continued fractions

Chowla, P. ; Chowla, S. (1972) Problems on periodic simple continued fractions Proceedings of the National Academy of Sciences of the United States of America, 69 (12). p. 3745. ISSN 0027-8424

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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 - a3-1 + −… ± 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 Hirzebruch (unpublished) recently found the surprising theorem (which is a special case of more general results): If p is a prime≡ 3(4) and p 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).

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