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\(\sqrt p \)L4p(1), where L4p(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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Dedicated to the 90th anniversary of G. M. Goluzin's birth
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 21–30.
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Golubeva, E.P. Estimates of the levy constant for\(\sqrt p \) and class number one criterion for ℚ(\(\sqrt p \)). J Math Sci 95, 2185–2191 (1999). https://doi.org/10.1007/BF02172462
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DOI: https://doi.org/10.1007/BF02172462