Related papers: Origin of the numerals, Zero concept
Back in 1755, Euler explored an interesting array of numbers that now frequently appears in polynomial identities, combinatorial problems, and finite calculus, among other places. These numbers share a strong connection with well-known…
We study some aspects of the emergence of logos from chaos on a basal model of the universe using methods and techniques from algorithmic information and Ramsey theories. Thereby an intrinsic and unusual mixture of meaningful and spurious,…
Given a sequence converging to zero, we consider the set of numbers which are sums of (infinite, finite, or empty) subsequences. When the original sequence is not absolutely summable, the subsum set is an unbounded closed interval which…
A new approach to the phenomenon of large numbers coincidence leads to unexpected results. No matter how strange it might sound, the exact value of cosmological parameters and their analytical expression through fundamental constants have…
Hilbert's Foundations of Geometry was perhaps one of the most influential works of geometry in the 20th century and its axiomatics was the first systematic attempt to clear up the logical gaps of the Elements. But does it have gaps of its…
Zero factorial, defined to be one, is often counterintuitive to students but nonetheless an interesting concept to convey in a classroom environment. The challenge is to delineate the concept in a simple and effective way through the…
Generalizing the concept of a perfect number, Sloane's sequences of integers A083207 lists the sequence of integers $n$ with the property: the positive factors of $n$ can be partitioned into two disjoint parts so that the sums of the two…
Students of quantum mechanics encounter discrete quantum numbers in a somewhat incoherent and bewildering number of ways. For each physical system studied, quantum numbers seem to be introduced in its own specific way, some enumerating from…
We introduce the binary value principle which is a simple subset-sum instance expressing that a natural number written in binary cannot be negative, relating it to central problems in proof and algebraic complexity. We prove conditional…
How many odd numbers are there? How many even numbers? From Galileo to Cantor, the suggestion was that there are the same number of odd, even and natural numbers, because all three sets can be mapped in one-one fashion to each other. This…
In this paper, we prove that the numerical-semigroup-gap counting problem is #NP-complete as a main theorem. A numerical semigroup is an additive semigroup over the set of all nonnegative integers. A gap of a numerical semigroup is defined…
Every quadratic form represents 0; therefore, if we take any number of quadratic forms and ask which integers are simultaneously represented by all members of the collection, we are guaranteed a nonempty set. But when is that set more than…
The twin primes conjecture is a very old problem. Tacitly it is supposed that the primes it deals with are finite. In the present paper we consider three problems that are not related to finite primes but deal with infinite integers. The…
If we form a decimal where the nth digit is the last non-zero digit of $n!$ (likewise, the last non-zero digit of $n^n$), we obtain an irrational number
Traditionally, semantics has been seen as a feature of human language. The advent of the information era has led to its widespread redefinition as an information feature. Contrary to this praxis, I define semantics as a special kind of…
The notion of stiffness, which originated in several applications of a different nature, has dominated the activities related to the numerical treatment of differential problems for the last fifty years. Contrary to what usually happens in…
The induction principle for natural numbers expresses that when a property holds for some natural number a and is hereditary, then it holds for all numbers greater than or equal to a. We present a similar principle for real numbers.
In the 16th century, Simon Stevin initiated a modern approach to decimal representation of measuring numbers, marking a transition from the discrete arithmetic practised by the Greeks to the arithmetic of the continuum taken for granted…
Let $b$ be a numeration base. A $b$-Niven number is one that is divisible by the sum of its base $b$ digits. We introduce high degree $b$-Niven numbers. These are $b$-Niven numbers that have a power greater than $1$ that is $b$-Niven…
The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These…