The class numbers of real quadratic fields of discriminant 4p

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, ... l_L satisfy at least L/2 linear relations. We also obtain a new lower bound for the fundamental unit ε_p of the field ?( $sqrt p$ ) for almost all p under consideration: ε_p > p³/log^1+δp for all p≥x with O(x/log^1+δx) possible exceptions (here δ>0 is an arbitrary constant), and an estimate for the mean value of the class number of ?( $sqrt p$ ) with respect to averaging over ε_p: $$sumlimits_{p equiv 3 (bmod 4), varepsilon _p leqslant x} {h(p) = O(x)}$$ . Bibliography: 11 titles.