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1995 | Buch

Propaedeutics Kapitel lesen Erstes Kapitel lesen

The Queen of Mathematics

An Introduction to Number Theory

verfasst von: W. S. Anglin

Verlag: Springer Netherlands

Buchreihe : Kluwer Texts in the Mathematical Sciences

Enthalten in: Professional Book Archive

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Über dieses Buch

Like other introductions to number theory, this one includes the usual curtsy to divisibility theory, the bow to congruence, and the little chat with quadratic reciprocity. It also includes proofs of results such as Lagrange's Four Square Theorem, the theorem behind Lucas's test for perfect numbers, the theorem that a regular n-gon is constructible just in case phi(n) is a power of 2, the fact that the circle cannot be squared, Dirichlet's theorem on primes in arithmetic progressions, the Prime Number Theorem, and Rademacher's partition theorem.
We have made the proofs of these theorems as elementary as possible.
Unique to The Queen of Mathematics are its presentations of the topic of palindromic simple continued fractions, an elementary solution of Lucas's square pyramid problem, Baker's solution for simultaneous Fermat equations, an elementary proof of Fermat's polygonal number conjecture, and the Lambek-Moser-Wild theorem.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Propaedeutics
Abstract
A natural number is one of the numbers 0, 1, 2, 3,.... Number Theory, as it is traditionally understood, is that branch of mathematics which studies the natural numbers. It includes ordinary arithmetic. For example, figuring out why long division works is a problem in Number Theory. As we shall see, Number Theory goes much further than this.
W. S. Anglin
Chapter 2. Simple Continued Fractions
Abstract
Simple continued fractions are a powerful mixture of analysis and algebra which is as important in contemporary Number Theory as it was in the work of Lagrange (1736–1813), who used these fractions to give completely general solutions to the Diophantine equations Ax+By = C and x2 − Ry2 = C. In this chapter we give Lagrange’s solution to the first equation, and, in Chapter 4, we give a solution very much like that of Lagrange to the second equation.
W. S. Anglin
Chapter 3. Congruence
Abstract
Carl Friedrich Gauss begins the Disquisitiones Arithmeticae (1801):
if a number a divides the difference of the numbers b and c, b and c are said to be congruent relative to a; if not, b and c are noncongruent. The number a is called the modulus. If the numbers b and c are congruent, each of them is called a residue of the other.
W. S. Anglin
Chapter 4. x2 − Ry2 = C
Abstract
In the first three chapters, we presented the Number Theory of Fermat, Lagrange, and Gauss (respectively). In this chapter, we present a new solution of the Diophantine equation x2 − Ry2 = C, and we present a new solution to a puzzle proposed by Edouard Lucas in 1875. We also establish Lucas’s test for perfect numbers, and, finally, look at some recent work of Alan Baker.
W. S. Anglin
Chapter 5. Classical Construction Problems
Abstract
The ancient Greeks searched for a way of using straightedge and compass to trisect an arbitrary angle, and to draw a segment of length \(\sqrt[3]{2}\). They also tried to’ square the circle’, that is, construct a segment of length \(\sqrt \pi \). Finally, they struggled to find straightedge and compass constructions for regular polygons with 7, 9, 11, 13, and 17 sides. In all this they failed, but it was not proved until the nineteenth century that the reason for their failure was that all these problems are insoluble — except one. In 1796 Gauss discovered a straightedge and compass construction for the regular 17-sided polygon. It was this discovery, the first advance on construction problems in 2000 years, that motivated Gauss to devote himself to mathematics.
W. S. Anglin
Chapter 6. The Polygonal Number Theorem
Abstract
A polygonal number is a nonnegative integer of the form
$$m\frac{{t^{\text{2}} - t}} {2} + t$$
where m is a positive integer, and t is a nonnegative integer. For example, when m = 1, we have the triangular numbers 0, 1, 3, 6, 10, 15, and so on. These are called triangular numbers because n pebbles can be arranged in the form of an isosceles right triangle just in case n has the form (t2 − t)/2 + t.
W. S. Anglin
Chapter 7. Analytic Number Theory
Abstract
In this chapter we draw on real and complex analysis to present four beautiful theorems. The first is P. Dirichlet’s theorem that there are infinitely many primes in any arithmetic progression
$$ a,\,a + b,\,a + 2b,\,a + 3b,\, \ldots $$
(assuming a and b are relatively prime). The second, due to J. Lambek, L. Moser, and R. Wild, gives the order of the number of primitive Pythagorean triangles with area less than n. The third is the Prime Number Theorem, first proved, independently, by J. Hadamard and C. J. de la Vallée Poussin.
W. S. Anglin
Backmatter
Metadaten
Titel
The Queen of Mathematics
verfasst von
W. S. Anglin
Copyright-Jahr
1995
Verlag
Springer Netherlands
Electronic ISBN
978-94-011-0285-8
Print ISBN
978-94-010-4126-3
DOI
https://doi.org/10.1007/978-94-011-0285-8