A numeration system associates a unique string, Ξ(n), with each positive integer n, where each string is over the same finite alphabet. Various digit counting statistics of Ξ(n) are of interest with respect to a numeration system. An example is the digital sum, which is the sum of the digits in the number. We present a unifying framework for deriving identities for the generating functions of such statistics in many of the more popular numeration systems. 1

Abstract: In this article we revisit the Fibonacci sequence and extend it in various directions. We contend this is a good topic to get people interested in recurrences and closed form representation. We discuss various Fibonacci related sequences and finally represent two third order recurrence relations in terms of binomial sums and hypergeometric functions.

Every non-negative integer n has at least one digital expansion n = ∑ ɛkFk, k≥2

The members of the Committee appointed to examine the

In this work we propose to take one step back in the use of double base number systems for elliptic curve point scalar multiplication. Using a modified version of Yao’s algorithm, we go back from the popular double base chain representation to a more general double base system. Instead of representing an integer k as ∑n i=1 2bi ti 3 where (bi) and (ti) are two decreasing sequences, we only set a maximum value for both of them. Then, we analyze the efficiency of our new method using different bases and optimal parameters. In particular, we propose for the first time a binary/Zeckendorf representation for integers, providing interesting results. Finally, we provide a comprehensive comparison to state-of-the-art methods, including a large variety of curve shapes and latest point addition formulae speed-ups. Keywords: Double-base number system, Zeckendorf representation, elliptic curve, point scalar multiplication, Yao’s algorithm.

This paper is an attempt to bring some theory on the top of some previously unproved experimental statements about the double-base number system (DBNS). We use results from diophantine approximation to address the problem of converting integers into DBNS. Although the material presented in this article is mainly theoretical, the proposed algorithm could lead to very efficient implementations. Keywords: Double base number system, Ostrowski’s numeration, continued fractions, diophantine approximation 1.

Abstract. We show that for integers k ≥ 2 and n≥3, the diameter of the Cayley graph of SLn(Z/kZ) associated to a standard two-element generating set, is at most a constant times n 2 lnk. This answers a question of A. Lubotzky concerning SLn(Fp) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing elements of