Quadratic irrationals with fixed period length in the continued fraction expansion

The class numbers of real quadratic fields of discriminant 4p

“…It then turns out that 11,... , IL satisfy at least L/2 linear relations (see Corollary 1 of Theorem 1), and IL+l (equal to n or n --1) is determined by these elements and one more integral parameter, the dependence on the latter being linear. The regularity shows up most clearly when (1) has the form V~ = [n, 11,21L-i,... , lL-1,2/1, IL+12/1, 9 9 9 , 2n], i.e., IL = 211,12 = 2lL-1,.... For example, for primes of the form p = 3132972 -18265947 + 26624259, and only for them, expansion (1) takes the form v/-p = [n, 1, 6,1, 2, 3, 2, n, 2, 3, 2,1, 6,1, 2n], where n = 5159 -1777 > 0.…”

“…for some integer 3'-In [1] it was shown that s -0 (rood 4)if p ~-7 (mod 8), and t: = 2 (rood 4) if p -3 (rood 8). follows that (-1) L = (-1) (p-3)/4.…”

“…The number n = [v/~ is polynomial with respect to L1,... , IL+l and linear with respect to one more free variable (the powers in li are computed in [1]). In this paper we find regularities in (1) in the case where d = p -3 (rood 4),p is a prime.…”

“…The number n = [v/~ is polynomial with respect to L1,... , IL+l and linear with respect to one more free variable (the powers in li are computed in [1]). In this paper we find regularities in (1) in the case where d = p -3 (rood 4),p is a prime. It then turns out that 11,... , IL satisfy at least L/2 linear relations (see Corollary 1 of Theorem 1), and IL+l (equal to n or n --1) is determined by these elements and one more integral parameter, the dependence on the latter being linear.…”

“…In [1] we wrote out all irrationalities of the form ~ with fixed even length of the period of the expansion into a continued fraction (cf. [2,3]).…”