Related papers: Euclid's Number-Theoretical Work

The pursue of what are properties that can be identified to permit an automated reasoning program to generate and find new and interesting theorems is an interesting research goal (pun intended). The automatic discovery of new theorems is a…

In his G\'eom\'etrie (1637) Descartes introduces the algebra of segments. This is a fundamental step in the mathematical treatment of variable quantities before the creation of differential calculus. It is an algebra with symbols but…

The statement of the Gauss-Bonnet theorem brings up an unexpected form of reflexivity (major concept of philosophy of mathematics), so that geometry contemplates itself in it. It is therefore the revolutionary and multifaceted concept of…

There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of…

The theory of equidistribution is about hundred years old, and has been developed primarily by number theorists and theoretical computer scientists. A motivated uninitiated peer could encounter difficulties perusing the literature, due to…

This is a translation from the Latin of Euler's "Problema algebraicum de inveniendis quatuor numeris ex datis totidem productis uniuscuiusque horum numerorum in summas trium reliquorum", Opera Postuma 1 (1862), 282-287, reprinted in…

E565 in the Enestrom index. Translated from the Latin original, "De plurimis quantitatibus transcendentibus quas nullo modo per formulas integrales exprimere licet" (1775). Euler does not prove any results in this paper. It seems to me like…

This book introduces the new research area of Geometric Data Science, where data can represent any real objects through geometric measurements. The first part of the book focuses on finite point sets. The most important result is a complete…

The proof of the theorem, which states that the Euclidean metric on the set of random points in an $n$-dimensional Euclidean space with the distribution of a special class, converges in probability in the limit $n\rightarrow\infty$ to the…

If several independent algorithms for a computer-calculated quantity exist, then one can expect their results (which differ because of numerical errors) to follow approximately Gaussian distribution. The mean of this distribution,…

Translated from the Latin original, "Theorema arithmeticum eiusque demonstratio", Commentationes arithmeticae collectae 2 (1849), 588-592. E794 in the Enestroem index. For m distinct numbers a,b,c,d,...,\upsilon,x this paper evaluates \[…

Translation from the Latin of Euler's "Observatio de summis divisorum" (1752). E243 in the Enestroem index. The pentagonal number theorem is that $\prod_{n=1}^\infty (1-x^n)=\sum_{n=-\infty}^\infty (-1)^n x^{n(3n-1)/2}$. This paper assumes…

Residues to a given modulus have been introduced to mathematics by Carl Friedrich Gauss with the definition of congruence in the `Disquisitiones Arithmeticae'. Their extraordinary properties provide the basis for a change of paradigm in…

Whilst Paul de Casteljau is now famous for his fundamental algorithm of curve and surface approximation, little is known about his other findings. This article offers an insight into his results in geometry, algebra and number theory.…

We give a Euclidean division algorithm for the real quadratic fields $\mathbb{Q}(\sqrt{m})$ for $m \in \{2, 3, 6, 7, 11, 19\}$, with the property that the norm of the remainder depends on the first Euclidean minimum of the field. In each…

Greatest Common Divisor (GCD) computation is one of the most important operation of algorithmic number theory. In this paper we present the algorithms for GCD computation of $n$ integers. We extend the Euclid's algorithm and binary GCD…

While the contents of Euclid's Elements are well-known these days, some characters of the original text have been overlooked due to interpretation by modern mathematical languages. The lens of modern mathematics once anachronistically…

In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has…

Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$, \begin{align*}…

Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method that Euler did not recommend, that Legendre…