Lengths of the periods of the continued fraction expansion of quadratic irrationalities and on the class numbers of real quadratic fields

The fundamental result of the paper is the following theorem: suppose that the Riemann conjecture is valid for the Dedekind ζ-functions of all fields Then there exists a constant C>0 such that on the interval p≤x one can find at least Cx log⁻¹ x prime numbers p for which h(5p²)=2. Here h(d) is the number of proper equivalence classes of primitive binary quadratic forms of discriminant d. In addition, it is proved that . For these sequence of discriminants of a special form with increasing square-free part, one has obtained a nontrivial estimate from above for the number of classes. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 72–81, 1987.     

Note on the class numbers of certain real quadratic fields

We prove that the class number of the real quadratic field is divisible byn forany integern ≥ 2 andany odd integera ≥ 3.     

Computations of class numbers of real quadratic fields

In this paper an unconditional probabilistic algorithm to compute the class number of a real quadratic field is presented, which computes the class number in expected time . The algorithm is a random version of Shanks' algorithm. One of the main steps in algorithms to compute the class number is the approximation of . Previous algorithms with the above running time , obtain an approximation for by assuming an appropriate extension of the Riemann Hypothesis. Our algorithm finds an appoximation for without assuming the Riemann Hypothesis, by using a new technique that we call the `Random Summation Technique'. As a result, we are able to compute the regulator deterministically in expected time . However, our estimate of on the running time of our algorithm to compute the class number is not effective.

     

About statistics of periods of continued fractions of quadratic irrationalities

In this paper we answer certain questions posed by V.I. Arnold, namely, we study periods of continued fractions for solutions of quadratic equations in the form x ²+px=q with integer p and q, p ²+q ²≤R ². We prove a weak variant of Arnold conjectures about the Gauss–Kuzmin statistics with R→∞.     

Quadratic equations over finite fields and class numbers of real quadratic fields

Letp be an odd prime and the finite field withp elements. In the present paper we shall investigate the number of points of certain quadratic hypersurfaces in the vector space and derive explicit formulas for them. In addition, we shall show that the class number of the real quadratic field (wherep1 (mod 4)) over the field of rational numbers can be expressed by means of these formulas.     

Simultaneous indivisibility of class numbers of pairs of real quadratic fields

The Ramanujan Journal - For a square-free integer t, Byeon (Proc. Am. Math. Soc. 132:3137–3140, 2004) proved the existence of infinitely many pairs of quadratic fields $$\mathbb...     

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: $$\sum\limits_{p \equiv 3 (\bmod 4), \varepsilon _p \leqslant x} {h(p) = O(x)}$$ . Bibliography: 11 titles.     

Fibonacci numbers and real quadratic p-rational fields

Periodica Mathematica Hungarica - We characterize p-rational real quadratic fields in terms of generalized Fibonacci numbers. We then use this characterization to give numerical evidence to a...     

Class numbers of quadratic function fields and continued fractions

A function field version of a theorem of F. Hirzebruch relating continued fractions to class numbers of quadratic number fields is established. Our approach is based on Artin's thesis and Zagier's proof of Hirzebruch's theorem. Some of our results seem to be of independent interest, e.g. explicit formulas for Zeta functions of real quadratic function fields.     

Quadratic irrationals with short periods of expansion into continued fraction

We study the class number of an indefinite binary quadratic form of discriminant d based on the expansion of d into a continued fraction and single out sequences of d for which h(d) has a lower-bound extimate. Progress is made for the conjecture on the estimate of the quantity of prime discriminants d with fixed length of period of expansion of d. Bibliography: 15 titles.Dedicated to the 90th anniversary of G. M. Goluzin's birthTranslated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 31–45.     

Congruence properties of class numbers of quadratic fields

The authors prove that the class number of the quadratic field Q(√?g) is divisible by 3 if g is a prime of the form 27n² + 4.     

Computation of class numbers of quadratic number fields

We explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields. We therefore end up with a simpler and faster method for computing class numbers of quadratic number fields. We also explain how to end up with a simpler and faster method for computing relative class numbers of imaginary abelian number fields.

     

On the exceptional fields for a class of real quadratic fields

In this paper, we give a lower bound exp(2.2 × 10~8 ) for those discriminants of real quadratic fields Q(√ d) with d= N~2-4 and h(d)=1.     

Computing the Hilbert class field of real quadratic fields

Using the units appearing in Stark's conjectures on the values of -functions at , we give a complete algorithm for computing an explicit generator of the Hilbert class field of a real quadratic field.

     

Distribution of the reduced quadratic irrationals arising from the odd continued fraction expansion

This paper investigates the quadratic irrationals that arise as periodic points of the Gauss type shift associated to the odd continued fraction expansion. It is shown that these numbers, which we call O-reduced, when ordered by the length of the associated closed primitive geodesic on some modular surface $\Gamma ?\mathbb{H}$, are equidistributed with respect to the Lebesgue absolutely continuous invariant probability measure of the Odd Gauss shift.