Related papers: Absolutely Abnormal Numbers

For an arbitrary integer $x$, an integer of the form $T(x)=\frac{x^2+x}{2}$ is called a triangular number. For positive integers $\alpha_1,\alpha_2,\dots,\alpha_k$, a sum…

Contrary to popular misconception, the question in the title is far from simple. It involves sets of numbers on the first level, sets of sets of numbers on the second level, and so on, endlessly. The infinite hierarchy of the levels…

In this paper we recall a non-standard construction of the Borel sigma-algebra B in [0,1] and construct a family of measures (in particular, Lebesgue measure) in B by a completely non-topological method. This approach, that goes back to…

In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.

In this work, we determined the general terms of all almost balancing numbers of first and second type in terms of balancing numbers and conversely we determined the general terms of all balancing numbers in terms of all almost balancing…

We study the convergence of certain subseries of the harmonic series corresponding to increasing sequences of integers whose digits in a certain base are not uniformly distributed. We also discuss the case of irregular sequences, where the…

We define a new condition number adapted to directionally uniform perturbations. The definitions and theorems can be applied to a large class of problems. We show the relation with the classical condition number, and study some interesting…

Much is known about asymptotic expansions for asymptotically normal distributions if these distributions are either absolutely continuous or pure lattice distributions. In this paper we begin an investigation of the discrete but non-lattice…

Much recent progress has been made concerning the probable existence of Odd Perfect Numbers, forming part of what has come to be known as Sylvester's Web Of Conditions. This paper proves some results concerning certain properties of the…

Some aspects of the development of physics and the mathematics set one think about relation between complex numbers and reality around us. If number to spot as the relation of two quantities, from the fact of existence of complex numbers…

A new kind of numbers called Hyper Space Complex Numbers and its algebras are defined and proved. It is with good properties as the classic Complex Numbers, such as expressed in coordinates, triangular and exponent forms and following the…

We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…

A curious number is a palindromic number whose base ten representation has the form $a \ldots a b \ldots b a \ldots a$. In this paper, we determine all curious numbers that are perfect squares. Our proof involves reducing the search for…

James Maynard has taken the analytic number theory world by storm in the last decade, proving several important and surprising theorems, resolving questions that had seemed far out of reach. He is perhaps best known for his work on small…

We give a completely normal element in the maximal real subfield of a cyclotomic field over the field of rational numbers, which is different from that of Okada. This result is a consequence of the criterion for a normal element developed…

Do scientific theories limit human knowledge? In other words, are there physical variables hidden by essence forever? We argue for negative answers and illustrate our point on chaotic classical dynamical systems. We emphasize parallels with…

Computing the embedding distribution of a given graph is a fundamental question in topological graph theory. In this article, we extend our viewpoint to a sequence of graphs and consider their asymptotic embedding distributions, which are…

Randomness comes in two qualitatively different forms. Apparent randomness can result both from ignorance or lack of control of degrees of freedom in the system. In contrast, intrinsic randomness should not be ascribable to any such cause.…

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.

In this work, we established symmetric representation of numbers where one can use any of 9 digits giving the same number. The representations of natural numbers from 0 to 1000 are given using only single digit in all the nine cases, i.e.,…