The primary purpose of this thesis is to give a substantial generalization of the set of integers Z, where particular emphasis is given to number theoretic questions such as that of unique factorization. The origin of the thesis came from a study of a special case of generalized integers called the Gaussian Integers, namely the set of all complex numbers in the form n + mi, for m,n in Z. The main generalization involves what are called algebraic integers.

<p>In this thesis it is investigated if representing a field <i>Z</i><i>p</i><i>, p</i> = 1 (mod 4) prime, by another field <i>Z[i]</i>/ < <i>a + bi </i>> over the gaussian integers, with <i>p</i> = <i>a</i><i>2</i><i> + b</i><i>2</i>, results in arithmetic architectures using a smaller number of logic gates. Only bit parallell architectures are considered and the programs Espresso and SIS are used for boolean minimization of the architectures. When counting gates only NAND, NOR and inverters are used.</p><p>Two arithmetic operations are investigated, addition and multiplication. For addition the architecture over<i> Z[i]/ < a+bi ></i> uses a significantly greater number of gates compared with an architecture over<i> Z</i><i>p</i>. For multiplication the architecture using gaussian integers uses a few less gates than the architecture over <i>Z</i><i>p</i> for <i>p</i> = 5 and for<i> p</i> = 17 and only a few more gates when <i>p</i> = 13. Only the values 5, 13, 17 have been compared for multiplication. For addition 12 values, ranging from 5 to 525313, have been compared.</p><p>It is also shown that using a blif model as input architecture to SIS yields much better performance, compared to a truth table architecture, when minimizing.</p>

In this thesis it is investigated if representing a field Zp, p = 1 (mod 4) prime, by another field Z[i]/ &lt; a + bi &gt; over the gaussian integers, with p = a2 + b2, results in arithmetic architectures using a smaller number of logic gates. Only bit parallell architectures are considered and the programs Espresso and SIS are used for boolean minimization of the architectures. When counting gates only NAND, NOR and inverters are used. Two arithmetic operations are investigated, addition and multiplication. For addition the architecture over Z[i]/ &lt; a+bi &gt; uses a significantly greater number of gates compared with an architecture over Zp. For multiplication the architecture using gaussian integers uses a few less gates than the architecture over Zp for p = 5 and for p = 17 and only a few more gates when p = 13. Only the values 5, 13, 17 have been compared for multiplication. For addition 12 values, ranging from 5 to 525313, have been compared. It is also shown that using a blif model as input architecture to SIS yields much better performance, compared to a truth table architecture, when minimizing.

Our main result is that there are infinitely many primes of the form a² + b² such that a² + 4b² has at most 5 prime factors. We prove this by first developing the theory of $L$-functions for Gaussian primes by using standard methods. We then give an exposition of the Siegel--Walfisz Theorem for Gaussian primes and a corresponding Prime Number Theorem for Gaussian Arithmetic Progressions. Finally, we prove the main result by using the developed theory together with Sieve Theory and specifically a weighted linear sieve result to bound the number of prime factors of a² + 4b². For the application of the sieve, we need to derive a specific version of the Bombieri--Vinogradov Theorem for Gaussian primes which, in turn, requires a suitable version of the Large Sieve. We are also able to get the number of prime factors of a² + 4b² as low as 3 if we assume the Generalised Riemann Hypothesis.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-28T13:01:20Z No. of bitstreams: 1 arquivototal.pdf: 553092 bytes, checksum: 60c2a1a060ead1662c0a4edc6ec82f9c (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2017-08-28T15:55:23Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 553092 bytes, checksum: 60c2a1a060ead1662c0a4edc6ec82f9c (MD5) / Made available in DSpace on 2017-08-28T15:55:23Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 553092 bytes, checksum: 60c2a1a060ead1662c0a4edc6ec82f9c (MD5) Previous issue date: 2015-10-23 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Our study aims to present a special category of numbers, the Gaussian integers, their properties and operations, have an overview about these numbers, their history and emergence. We will also study Gaussian prime numbers, their properties and application in matrix language representation of 2 x 2 type. / Nosso estudo tem como objetivo apresentar uma categoria especial de números, os inteiros Gaussianos, suas propriedades e operações, ter uma visão geral sobre esses números, sua história e surgimento. Também estudaremos números primos Gaussianos, suas propriedades e aplicação com representação em linguagem matricial do tipo 2 x 2.

Nous construisons, pour les corps quadratiques imaginaires avec nombre de classes 1, un site arithmétique de type Connes-Consani. La principale difficulté ici est que les constructions de Connes et Consani et une partie de leurs résultats reposent sur la relation d'ordre naturellement présente sur les nombres réels qui est compatible avec les opérations arithmétiques basiques. Bien sûr rien de la sorte n'existe pas dans le cas des corps quadratiques imaginaires avec nombre de classes 1. Nous définissons ce que nous appelons le site arithmétique pour de tels corps de nombres, puis nous calculons les points de ces sites arithmétiques et nous les exprimons en termes de l'espace des classes d'adèles considéré par Connes pour donner une interprétation spectrale des zéros des fonctions L de Hecke. On obtient alors que pour un corps quadratique imaginaire avec nombre de classes 1, les points de notre site arithmétique sont reliés aux zéros de la fonction zêta de Dedekind du corps de nombres considéré et aux zéros de certaines fonctions L de Hecke. Nous étudions ensuite la relation entre le spectre de l'anneau des entiers du corps de nombres et le site arithmétique. Enfin nous construisons le carré du site arithmétique. / We construct, for imaginary quadratic number fields with class number 1, an arithmetic site of Connes-Consani type. The main difficulty here is that the constructions of Connes and Consani and part of their results strongly rely on the natural order existing on real numbers which is compatible with basic arithmetic operations. Of course nothing of this sort exists in the case of imaginary quadratic number fields with class number 1. We first define what we call arithmetic site for such number fields, we then calculate the points of those arithmetic sites and we express them in terms of the ad\`eles class space considered by Connes to give a spectral interpretation of zeroes of Hecke L functions of number fields. We get therefore that for a fixed imaginary quadratic number field with class number 1, that the points of our arithmetic site are related to the zeroes of the Dedekind zeta function of the number field considered and to the zeroes of some Hecke L functions. We then study the relation between the spectrum of the ring of integers of the number field and the arithmetic site. Finally we construct the square of the arithmetic site.

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