Related papers: Logarithm of Irrationals and Beatty Sequences

We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (a, a^2) or (a,…

Numbers are often used to define more complicated numbers. For example, two integers are used to define a rational number and two reals are used to define a complex number. It might be expected that an irrational power of an irrational…

We derive an identity connecting any two second-order linear recurrence sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are…

The sums of three trigonometric series with logarithmic coefficients are derived by extending an approach first utilized by Lerch. By applying Frullani's theorem to two of these series, two non-trivial integrals involving hyperbolic…

We present a systematic study of integrals over [0,1] where the integrand is of the form Q(x) log log 1/x. Here Q is a rational function.

For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…

We prove that a real number a greater than or equal to 2 is the irrationality exponent of some computable real number if and only if a is the upper limit of a computable sequence of rational numbers. Thus, there are computable real numbers…

We show that every irrational number is a sum of two real numbers with diverging partial quotients. The proof is constructive. The key towards these results is an algorithm which was recently developed by Nikita Shulga, and our study of…

In the literature, we have various ways of proving irrationality of a real number. In this survey article, we shall emphasize on a particular criterion to prove irrationality. This is called nice approximation of a number by a sequence of…

Two general methods for establishing the logarithmic behavior of recursively defined sequences of real numbers are presented. One is the interlacing method, and the other one is based on calculus. Both methods are used to prove logarithmic…

In this paper we express the difference of two complementary Beatty sequences, as the sum of two Beatty sequences closely related to them. In the process we introduce a new Algorithm that generalizes the well known Minimum Excluded…

We study logarithmic integrals of the form $\int_0^1 x^i\ln^n(x)\ln^m(1-x)dx$. They are expressed as a rational linear combination of certain rational numbers $(n,m)_i$, which we call tiered binomial coefficients, and products of the zeta…

This paper investigates the Stirling numbers of the first and second kind associated with a delta series f (t). These numbers provide a robust framework that satisfies the orthogonality and inverse relations, often lacking in recent…

Let $n$ be a positive integer. In this paper we estimate the size of the set of linear forms $b_1\log a_1 + b_2\log a_2+...+b_n\log a_n$, where $|b_i|\leq B_i$ and $1\leq a_i\leq A_i$ are integers, as $A_i,B_i\to \infty$.

We characterize all the pairs of complementary non-homogenous Beatty sequences $(A_n)_{n\ge 0}$ and $(B_n)_{n\ge 0}$ for which there exists an invariant game having exactly $\{(A_n,B_n)\mid n\ge 0\}\cup \{(B_n,A_n)\mid n\ge 0\}$ as set of…

In the paper, by establishing a new and explicit formula for computing the $n$-th derivative of the reciprocal of the logarithmic function, the author presents new and explicit formulas for calculating Bernoulli numbers of the second kind…

Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow…

We establish that the sequences formed by logarithms and by "fractional" powers of integers, as well as the sequence of prime numbers, are non-holonomic, thereby answering three open problems of Gerhold [Electronic Journal of Combinatorics…

In this paper we present some results related to the problem of finding periodic representations for algebraic numbers. In particular, we analyze the problem for cubic irrationalities. We show an interesting relationship between the…

The Fibonacci sequence is a series of positive integers in which, starting from $0$ and $1$, every number is the sum of two previous numbers, and the limiting ratio of any two consecutive numbers of this sequence is called the golden ratio.…