相关论文: Beyond Complex Numbers

Number sequences defined by a linear recursion relation are studied by means of generating functions. Indices of the terms in the recursion relation have arbitrary differenses. In addition to formulas for the nth term an algorithm is…

We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also…

We consider the rooted trees which not have isomorphic representation and introduce a conception of complexity a natural number also. The connection between quantity such trees with $n$ edges and a complexity of natural number $n$ is…

An odd prime $p$ is called irregular with respect to Euler polynomials if it divides the numerator of one of the numbers $$E_1(0),E_{3}(0),\ldots,E_{p-2}(0),$$ where $E_n(x)$ is the $n$-th Euler polynomial. As in the classical case, we link…

We give a combinatorial model for the bounded derived category of graded modules over the dual numbers in terms of arcs on the integer line with a point at infinity. Using this model we describe the lattice of thick subcategories of the…

Computational complexity is a core theory of computer science, which dictates the degree of difficulty of computation. There are many problems with high complexity that we have to deal, which is especially true for AI. This raises a big…

Resolving a conjecture of Zhi-Wei Sun, we prove that every rational number can be represented as a sum of distinct unit fractions whose denominators are practical numbers. The same method applies to allowed denominators that are closed…

We first give a summary of the history of transcendental numbers then use a nice technique by G. Dresden to prove a new transcendental number. In particular, while previous work looked at the last non-zero digit of $n^n$, we consider the…

The $abc$ conjecture is a very deep concept in number theory with wide application to many areas of number theory. In this article we introduce the conjecture and give examples of its applications. In particular we apply the $abc$…

We can generalize the definition of {\it splitting number } $s(\kappa )$ for $\kappa$ uncountable regular: $s(\kappa )=min\{ |\Cal S|:\Cal S\subset \Cal P(\kappa ) \forall a\in \kappa ^\kappa \exists b\in \Cal S |a\cap b|=|a\setminus…

In this paper, we study the sum of the divisor function over sets with digit restrictions.

The decimal digits of $\pi$ are widely believed to behave like as statistically independent random variables taking the values $0, 1, 2, 3, 4, 5$, $6, 7, 8, 9$ with equal probabilities $1/10$. In this article, first, another similar…

Algebraic integers in totally imaginary quartic number fields are not discrete in the complex plane under a fixed embedding, which makes it impossible to visualize all integers in the plane, unlike the quadratic imaginary algebraic…

This paper investigates the length of the repeating decimal part when a fraction is expressed in decimal form. First, it provides a detailed explanation of how to calculate the length of the repeating decimal when the denominator of the…

In considering the reliability of numerical programs, it is normal to "limit our study to the semantics dealing with numerical precision" (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that…

We show that the nonlinear real arithmetic theory (NRA) as defined in the SMTLIB standard is undecidable. The undecidability arises from the treatment of division by zero as an uninterpreted function, which allows encoding integer…

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

In this book for the first time the authors introduce the notion of real neutrosophic complex numbers. Further the new notion of finite complex modulo integers is defined. For every $C(Z_n)$ the complex modulo integer $i_F$ is such that…

Treating differentials as independent algebraic units have a long history of use and abuse. It is generally considered problematic to treat the derivative as a fraction of differentials rather than as a holistic unit acting as a limit,…

Every natural number greater than $2$ can be written as the sum of a prime and a square-free number, and recent work has imposed additional divisibility conditions on the square-free number. We overcome limitations in these works to prove…