Editor's Gloss: Counting the Numbers

Counting Keith numbers

A Keith number is a positive integer N with the decimal representation a1a2 . . . an such that n ≥ 2 and N appears in the sequence (Km)m≥1 given by the recurrence K1 = a1, . . . ,Kn = an and Km = Km−1 + Km−2 + · · · + Km−n for m > n. We prove that there are only finitely many Keith numbers using only one decimal digit (i.e., a1 = a2 = · · · = an), and that the set of Keith numbers is of asympto...

Numbers and Counting

Counting Square-Free Numbers

The main topic of this contribution is the problem of counting square-free numbers not exceeding n. Before this work we were able to do it in time Õ( √ n). Here, the algorithm with time complexity Õ(n) and with memory complexity Õ(n) is presented. Additionally, a parallel version is shown, which achieves full scalability. As of now the highest computed value was for n = 10. Using our implementa...

The Effects of Counting the Stride Numbers on the Parkinsonian Gait: Suggesting a Possible Reason for Dual Task Interference

Introduction: Parkinson Disease (PD) is a degenerative and progressive disorder of the central nervous system. It results from degeneration of Substantia Nigra pars compacta (SNc) of the Basal Ganglia (BG). Gait disturbances in PD patients generally get worse over time. The underlying mechanism of gait disturbances in PD has not been elucidated yet. Methods: In this study, we tried to analyze ...

Counting Perron Numbers by Absolute Value

We count various classes of algebraic integers of fixed degree by their largest absolute value. The classes of integers considered include all algebraic integers, Perron numbers, totally real integers, and totally complex integers. We give qualitative and quantitative results concerning the distribution of Perron numbers, answering in part a question of W. Thurston [Thu].