ISSN:
1573-8795
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract For p prime, p≡3 (mod 4), we study the expansion of $$\sqrt p$$ into a continued fraction. In particular, we show that in the expansion $$\sqrt p = [n,\overline {l_1 ,...,l_L ,l,L_L ,...,l_1 ,2n} ]$$ l1, ... lL satisfy at least L/2 linear relations. We also obtain a new lower bound for the fundamental unit εp of the field ℚ( $$\sqrt p$$ ) for almost all p under consideration: εp 〉 p3/log1+δp for all p≥x with O(x/log1+δx) possible exceptions (here δ〉0 is an arbitrary constant), and an estimate for the mean value of the class number of ℚ( $$\sqrt p$$ ) with respect to averaging over εp: $$\sum\limits_{p \equiv 3 (\bmod 4), \varepsilon _p \leqslant x} {h(p) = O(x)}$$ . Bibliography: 11 titles.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02366457
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