The parity of the period of the continued fraction of d

Abstract

Abstract. We prove that, asymptotically, in the set of squarefree integers d, not divisible by primes congruent to 3 mod 4, the period of the expansion of √ d in continued fractions is more frequently odd than even. Statement of the resultsThe subject of the expansion of the real numbers in simple continued fractions remains a very opaque domain in the theory of numbers. One of the very few achievements of this theory is the following famous theorem due to Lagrange (see [22, Theorem 3 p. 317], for instance). With the classical notations, we have the equalityIn that formula, s = s(d) is the period of the expansion, and we have the equality a t = a s−t , for any 1 ≤ t ≤ s − 1 and also the inequality s < 2d. Hence Theorem A defines an application s : d → s(d), from the set of non square integers to N * . The image of s is equal to N * (see [22, Theorem 6 p.325]) and more precisely for every positive integer s 0 , the equation s(d) = s 0 has infinitely many solutions in d.Recall that if the real number α has an ultimately periodic expansion in continued fractions, then α is an algebraic number of degree 2 (see [22, p.328] for instance).The application s is very mysterious at many points of view. Here we shall be concerned by the frequency of the odd values of this function. We recall a very useful link between the parity of s(d) and the associated negative Pell equation.

Weighted Distribution of the 4-rank of Class Groups and Applications

“…We shall closely follow the proofs given in [6], [8] & [9] and only quote the parts which have to be modified in order to incorporate the weight κ ω(|D|) . To shorten this part, we shall even restrict ourselves to the case of odd negative fundamental D or of odd (positive) special D. As usual the prime 2 creates extra difficulty which is only of practical order, but not of theoretical one (the same phenomenon already appeared in [6], [8] & [9]). In §4, we shall develop some applications of Corollary 1 to some questions raised in [24].…”

“…As said above, Theorems 1 and 2 are only variations of results contained in [6], [8] and [9]. Our aim is to enumerate the modifications of the original proofs to incorporate the coefficient κ ω(|D|) .…”

“…Hence following the proofs of [6,Theorems 7,8,9,10 & 11], [8,Theorems 3 & 4] and [9, Theorem 3], we obtain the evaluation of the corresponding weighted moments. To state our results in a global way, we introduce the following moments…”

The infinitude of $\mathbb {Q}(\sqrt {-p})$ with class number divisible by 16

“…In terms of a and b from (7), this means that 8|h ⇐⇒ a + b ≡ ±1 mod 8. We remark that Fouvry and Klüners developed similar methods in [5], where they constructed an analogue of the 4-Hilbert class field to deduce a criterion for the 8-rank of class groups in a family of real quadratic number fields. From now on, suppose that 8|h.…”

“…so that U 2 = −1 × U (5) . In other words, u ∈ U is a square in Q 2 (i) if and only if u ≡ ±1 (mod m 5 ).…”