 By Pierre Samuel

Best number theory books

A Course In Algebraic Number Theory

This can be a textual content for a simple direction in algebraic quantity idea.

Reciprocity Laws: From Euler to Eisenstein

This ebook is set the advance of reciprocity legislation, ranging from conjectures of Euler and discussing the contributions of Legendre, Gauss, Dirichlet, Jacobi, and Eisenstein. Readers a professional in simple algebraic quantity concept and Galois conception will locate targeted discussions of the reciprocity legislation for quadratic, cubic, quartic, sextic and octic residues, rational reciprocity legislation, and Eisenstein's reciprocity legislations.

Einführung in die Wahrscheinlichkeitstheorie und Statistik

Dieses Buch wendet sich an alle, die - ausgestattet mit Grundkenntnissen der Differential- und Intergralrechnung und der linearen Algebra - in die Ideenwelt der Stochastik eindringen möchten. Stochastik ist die Mathematik des Zufalls. Sie ist von größter Bedeutung für die Berufspraxis der Mathematiker.

Einführung in Algebra und Zahlentheorie

Das Buch bietet eine neue Stoffzusammenstellung, die elementare Themen aus der Algebra und der Zahlentheorie verknüpft und für die Verwendung in Bachelorstudiengängen und modularisierten Lehramtsstudiengängen konzipiert ist. Es führt die abstrakten Konzepte der Algebra in stetem Kontakt mit konkreten Problemen der elementaren Zahlentheorie und mit Blick auf Anwendungen ein und bietet Ausblicke auf fortgeschrittene Themen.

Additional resources for Algebraic theory of numbers

Sample text

See Dickson’s conjecture; Dirichlet; Hardy-Littlewood conjectures Aurifeuillian factorization Since a2 + b2 cannot be factorized into two algebraic factors, unlike a2 − b2 = (a + b)(a − b), we might assume that n4 + 1, which is also the sum of two squares, cannot be factorized. Not so! qxd 3/22/05 12:05 PM Page 15 average prime • 15 ෆ+ Now we can see a connection: a2 + b2 = (a + b)2 − 4ab = (a − ͙ab ෆ + b), which normally “doesn’t count” because of the b)(a + ͙ab square roots. It follows that n4 + 1 is always composite, except when n2 − n + 1 = 1 and n = 0 or 1.

Dirichlet proved in 1837, a conjecture made by Gauss: if a and b are coprime positive integers, then the arithmetic progression a, a + b, a + 2b, a + 3b, . . contains infinitely many primes. He did so by proving that if p is a prime of the form an + b, with a and b coprime, then the sum of all the primes p of this form less than x is approximately, 1 ᎏ и log log x φ(a) as x tends to infinity. In other words, it increases without limit, albeit very slowly, and so the primes of that form cannot be finite in number.

There are no primes among the first 13,500 terms. (Weisstein, MathWorld) consecutive numbers Sylvester proved in 1892 that every product of n consecutive integers greater than n is divisible by a prime greater than n. In fact, although the product of five consecutive integers 6 и 7 и 8 и 9 и 10 is divisible by just one prime, 7, greater than 5, the product 200 и 201 и 202 и 203 и 204 is divisible by five primes, 67, 101, 7, 29, and 17. This suggests that Sylvester’s result is rather generous: indeed, the product of just two consecutive numbers is always divisible by a prime greater than N, if the product is large enough.