Related papers: Computable Absolutely Pisot Normal Numbers

For scientific computations on a digital computer the set of real number is usually approximated by a finite set F of "floating-point" numbers. We compare the numerical accuracy possible with difference choices of F having approximately the…

Let $Q=(q_n)_{n=1}^\infty$ be a sequence of bases with $q_i\ge 2$. In the case when the $q_i$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$-Cantor series expansion is both…

We give a construction of an absolutely normal real number $x$ such that for every integer $b $ greater than or equal to $2$, the discrepancy of the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 0}$ is of asymptotic order…

In this expository article, the real numbers are defined as infinite decimals. After defining an ordering relation and the arithmetic operations, it is shown that the set of real numbers is a complete ordered field. It is further shown that…

We consider the following "partition and sum" operation on a natural number: Treating the number as a long string of digits insert several plus signs in between some of the digits and carry out the indicated sum. This results in a smaller…

We construct new families of completely regular codes by concatenation methods. By combining parity check matrices of cyclic Hamming codes, we obtain families of completely regular codes. In all cases, we compute the intersection array of…

The aim of this note is to show that any even perfect number, other than $6$, can be written as the sum of 5 cubes of natural numbers. We also conjecture that any even perfect number, other than $6$, can be written as the sum of only 3…

We construct the base $2$ expansion of an absolutely normal real number $x$ so that, for every integer $b$ greater than or equal to $2$, the discrepancy modulo $1$ of the sequence $(b^0 x, b^1 x, b^2 x , \ldots)$ is essentially the same as…

Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…

Defined by Borel, a real number is normal to an integer base $b$, greater than or equal to $2$, if in its base-$b$ expansion every block of digits occurs with the same limiting frequency as every other block of the same length. We consider…

An algorithm for computing /pi(N) is presented.It is shown that using a symmetry of natural numbers we can easily compute /pi(N).This method relies on the fact that counting the number of odd composites not exceeding N suffices to calculate…

We introduce a non-wellfounded proof system for intuitionistic logic extended with inductive and co-inductive definitions, based on a syntax in which fixpoint formulas are annotated with explicit variables for ordinals. We explore the…

We use representations and differentiation algorithms of posets, in order to obtain results concerning unsolved problems on figurate numbers. In particular, we present criteria for natural numbers which are the sum of three octahedral…

Normal numbers were introduced by Borel and later proven to be a weak notion of algorithmic randomness. We introduce here a natural relativization of normality based on generalized number representation systems. We explore the concepts of…

This note examines the infinite divisibility of density-based transformations of normal random variables. We characterize a class of density-based transformations of normal variables which produces non-infinitely divisible distributions. We…

We construct an absolutely normal number whose continued fraction expansion is normal in the sense that it contains all finite patterns of partial quotients with the expected asymptotic frequency as given by the Gauss-Kuzmin measure. The…

Let $g \geq 2$. A real number is said to be g-normal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let \phi denote Euler's totient function, let \sigma be the sum-of-divisors…

The Operator axioms have produced new real numbers with new operators. New operators naturally produce new equations and thus extend the traditional mathematical models which are selected to describe various scientific rules. So new…

We investigate the problem of finding integers $k$ such that appending any number of copies of the base-ten digit $d$ to $k$ yields a composite number. In particular, we prove that there exist infinitely many integers coprime to all digits…

We discuss various universality aspects of numerical computations using standard algorithms. These aspects include empirical observations and rigorous results. We also make various speculations about computation in a broader sense.