Artículos con la etiqueta ‘Number Theory (math.NT)’

Fray Juan de Ortega’s approximations, 500 years after

Por • 9 dic, 2012 • Category: sociologia

In 1512, on December 30, the first edition of Fray Juan de Ortega’s Arithmetic was published in Lyon. The last chapter, titled «Rules of Geometry», deals with lower approximations of 14 square roots. In later editions of the Arithmetic on 1534, 1537 and 1542 in Seville, these values are replaced by upper approximations. Twelve of them verify the Pell’s equation, and so they are optimal. At this moment nobody knows the way they were obtained. In this paper we show how these approximations can be obtained through a method consistent with the mathematical knowledge at that time.

Residues : The gateway to higher arithmetic I

Por • 1 dic, 2012 • Category: Educacion

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 arithmetic. By restricting residues to remainders left over by divison Peter Gustav Lejeune Dirichlet – Gauss’s successor in G\»ottingen – eliminated in his `Lectures on number theory’ the fertile concept of residues and attributed with the number-theoretic approach to residues for more than one and a half centuries to obscure Gauss’s paradigm shift in mathematics from elementary to higher arithmetic.

Self-Similarity in Geometry, Algebra and Arithmetic

Por • 27 nov, 2012 • Category: Crítica

We define the concept of self-similarity of an object by considering endomorphisms of the object as `similarity’ maps. A variety of interesting examples of self-similar objects in geometry, algebra and arithmetic are introduced. Self-similar objects provide a framework in which, one can unite some results and conjectures in different mathematical frameworks. In some general situations, one can define a well-behaved notion of dimension for self-similar objects. Morphisms between self-similar objects are also defined and a categorical treatment of this concept is provided. We conclude by some philosophical remarks.

On arithmetic numbers

Por • 30 jun, 2012 • Category: Ambiente

An integer $n$ is said to be \textit{arithmetic} if the arithmetic mean of its divisors is an integer. In this paper, using properties of the factorization of values of cyclotomic polynomials, we characterize arithmetic numbers. As an application, in Section 2, we give an interesting characterization of Mersenne numbers.