We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base b. We show that all of our so-called type III continued logarithms converge and all rational numbers have fin...

For the integer D = pq of the product of two distinct odd primes, we construct an elliptic curve E2rD : y 2 = x3 − 2rDx over Q, where r is a parameter dependent on the classes of p and q modulo 8, and show, under Birch and Swinnerton-Dyer conjecture, that the elliptic curve has rank one and vp(x([k]Q)) 6= vq(x([k]Q)) for odd k and a generator Q of the free part of E2rD(Q). Thus we can get p or ...

The distribution of the stellar and gaseous components in low surface brightness galaxies has been determined directly from optical and HI imaging. The distribution of what might be the dominant mass component, the dark matter, which is inferred from rotation curves, is far harder to determine. Although the rotation curves themselves can be determined fairly accurately from HI synthesis observa...

Let Fq be a finite field and let b and N be integers. We study the probability that the number of points on a randomly chosen elliptic curve E over Fq equals b modulo N . We prove explicit formulas for the cases gcd(N, q) = 1 and N = char(Fq). In the former case, these formulas follow from a random matrix theorem for Frobenius acting on the Ntorsion part of E, obtained by applying density resul...

In this paper we propose three refinements to Miller’s algorithm for computing Weil/Tate Pairing. The first one is an overall improvement and achieves its optimal behavior if the binary expansion of the involved integer has more zeros. If more ones are presented in the binary expansion, second improvement is suggested. The third one is especially efficient in the case base three. We also have s...

In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary Zp-extension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi [8] and Perrin-Riou [16], we define restricted Selmer groups and λ ± , µ ±-invariants; we then derive asymptotic formulas describing the growth of the Selmer group in ter...

Kolmogorov suggested to measure quality of a statistical hypothesis (a model) P for a data x by two parameters: Kolmogorov complexity C(P ) of the hypothesis and the probability P (x) of x with respect to P . The first parameter measures how simple the hypothesis P is and the second one how it fits. The paper [2] discovered a small class of models that are universal in the following sense. Each...

In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary Zp-extension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi [9] and Perrin-Riou [17], we define restricted Selmer groups and λ±, μ±-invariants; we then derive asymptotic formulas describing the growth of the Selmer group in terms ...