# Estimates of the levy constant for\(\sqrt p \) and class number one criterion for ℚ(\(\sqrt p \))

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## Abstract

Let p ≡ = 3 (mod 4) be a prime, let ℓ(\(\sqrt p \)) be the length of the period of the expansion of\(\sqrt p \) into a continued fraction, and let h(4p) be the class number of the field ℚ(\(\sqrt p \)). Our main result is as follows. For p > 91, h(4p)=1 if and only if ℓ(\(\sqrt p \)) > 0.56\(\sqrt p \)L_{4p}(1), where L_{4p}(1) is the corresponding Dirichlet series. The proof is based on studying linear relations between convergents of the expansion of\(\sqrt p \) into a continued fraction. Bibliography: 13 titles.

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© Kluwer Academic/Plenum Publishers 1999