# Elementary Number Theory

@inproceedings{Jones1976ElementaryNT, title={Elementary Number Theory}, author={Gareth A. Jones}, year={1976} }

Some preliminary considerations divisibility theory in the integers primes and their distribution the theory of congruences Fermat's theorem number-theoretic functions Euler's generalization of Fermat's theorem primitive roots and indices the quadratic reciprocity law perfect numbers the Fermat conjecture representation of integers as sums of squares Fibonacci numbers continued fractions some 20th-century developments appendices.

#### 443 Citations

A Comprehensive Course in Number Theory

- Mathematics
- 2012

Preface Introduction 1. Divisibility 2. Arithmetical functions 3. Congruences 4. Quadratic residues 5. Quadratic forms 6. Diophantine approximation 7. Quadratic fields 8. Diophantine equations 9.… Expand

Quadratic Reciprocity: Proofs and Applications

- Mathematics
- 2019

The law of quadratic reciprocity is an important result in number theory. The purpose of this thesis is to present several proofs as well as applications of the law of quadratic reciprocity. I will… Expand

On Giuga’s conjecture

- Mathematics
- 1995

In this paper we shall investigate Giuga’s conjecture which asserts an interesting characterization of prime numbers, just as Wilson’s Theorem. Some variations and consequences of the Giuga… Expand

Quadratic reciprocity for the rational integers and the Gaussian integers

- Mathematics
- 2010

This thesis begins by giving a brief time line of the origins of Number Theory. It highlights the big theorems that have been constructed in this subject, along with the mathematicians who… Expand

Some Divisibility Properties in Ring of Polynomials over a Unique Factorization Domain

- Mathematics
- 2008

Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the… Expand

An equivalence of Ward’s bound and its application

- Mathematics, Computer Science
- Des. Codes Cryptogr.
- 2011

This paper proves an equivalent condition of Ward's bound on dimension of divisible codes, which is part of a set of congruences having integer solutions, which makes the generalization of Ward’s bound an explicit one. Expand

On the Computation of Representations of Primes as Sums of Four Squares

- Mathematics
- 2007

Lagrange proved that every positive integer is the sum of four squares of natural numbers. Although Lagrange’s proof is constructive, it is not known whether the relative algorithm produces a… Expand

Fields with indecomposable multiplicative groups

- Mathematics
- 2016

Abstract We classify all finite fields and all infinite fields of characteristic not equal to 2 whose multiplicative groups are direct-sum indecomposable. For finite fields, we obtain our… Expand

On the quartic Gauss sums and their recurrence property

- Mathematics
- 2017

The main purpose of this paper is, using the method of trigonometric sums and the properties of Gauss sums, to study the computational problem of one kind of congruence equation modulo an odd prime… Expand

SOLUTIONS OF THE CUBIC FERMAT EQUATION IN QUADRATIC FIELDS

- Mathematics
- 2013

We give necessary and sufficient conditions on a squarefree integer d for there to be non-trivial solutions to x3 + y3 = z3 in , conditional on the Birch and Swinnerton-Dyer conjecture. These… Expand

#### References

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THE arithmetical questions treated by Diophantus of Alexandria, who flourished about the year 250 A.D., included such problems as the solution of the equationsHistory of the Theory of Numbers.By… Expand

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THIS book must be welcomed most warmly into X the select class of Oxford books on pure mathematics which have reached a second edition. It obviously appeals to a large class of mathematical readers.… Expand

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THE quaint words addressed “to the great variety of readers” by the editors of the folio Shakespeare of 1623 are equally applicable to the useful compendium of mathematical history which is the… Expand

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The wonderful achievement of Greek mathematics is here illustrated in two volumes of selected mathematical works. Volume I ("Loeb Classical Library no. 335") contains: The divisions of mathematics;… Expand

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WERE this book only for the mathematician it would be no book for me; but it is a great deal more. It is for all who care for the historical aspect of science; it is for all lovers of Greek, for… Expand

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I AM unable to find the passage in his works, but I think it was Prof. Ostwald who pointed out that while Aristotle was much more impressed with the retarding effect on the velocity of the mass of… Expand