Algunos problemas diofánticos

Abstract

The aim of this PhD thesis is the study of some diophantine problems. Some of them are related with Pythagorean triples and the others with arithmetical functions. In particular, chapter 1 begins with two historical notes about Diophantus' Arithmetic and the Plimpton tablet 322, to be followed, in chapter 2, by the study of how many Pythagorean triples are with legs less than n. Part of this chapter is contained in the article: Pythagorean triangles with legs less than n, published in the Journal of Computational and Applied Mathematics [7]. In chapter 3, we study the function (...). Mertens guessed that n. Odlyzco and te Riele demonstrated in [35] that the conjecture is false but without giving a counterexample. At present, it is a computational challenge to find an explicit counterexample of Mertens's conjecture. Odlyzco and te Riele predict there is not a counterexample less than 10(20). In chapter 4, we define functions similar to (...), and but, instead of taking integer values, they take gaussian integer values. These functions are based on one of the recurrent formula established in chapter 3. We will establish some recurrent formulas and bounds for these new functions. Chapter 5 collects part of the work that we have been doing for seven years about aliquot sequences. In the article: Advances in aliquot sequences published in January 1999 (vol. 68, num. 255) in Mathematics of Computation [4], we announced the advances achieved until July 1997. Later advances are shown in [5] and [6]. The most impressive result that we have obtained is to show that the aliquot sequence for the number 3630 stops at one after reach a 100-digit number. Published in Experimental Mathematics [3].