An Introduction to the Theory of Numbers - download pdf or read online

Posted by

By G. H. & Wright, E. M. Hardy

Show description

Read or Download An Introduction to the Theory of Numbers PDF

Similar number theory books

Get Number Theory and Its Applications PDF

"Addresses modern advancements in quantity idea and coding thought, initially offered as lectures at summer season university held at Bilkent collage, Ankara, Turkey. contains many ends up in booklet shape for the 1st time. "

Get The Riemann Hypothesis for Function Fields: Frobenius Flow PDF

This publication offers a lucid exposition of the connections among non-commutative geometry and the recognized Riemann speculation, targeting the idea of one-dimensional kinds over a finite box. The reader will stumble upon many very important facets of the idea, comparable to Bombieri's facts of the Riemann speculation for functionality fields, besides a proof of the connections with Nevanlinna thought and non-commutative geometry.

Additional info for An Introduction to the Theory of Numbers

Example text

Then (2) and xz= y (p) i mply NO for some integer k. Si nce lxl < JP and I Y I < posit i ve integer, k 1 and = Jp, x2 + y 2 < 2p so k < 2 . Si nce k i s a ( 3) ns desired. This is o ur first enco unter with a member of the c la ss of Diophantine equation s We shall hav e a lot to sa y a bo ut t his e qua tio n for some sma ll va lue s o f l m l nnu arbitrary n. 32 Polynomials We close this section by studying the Diophantine equation x2 _ yz = p. (4) + The seemingly innocuous change of sign makes the problem trivial.

THE GRO UP (f)(n) We are now ready to complete our study of (n). We shall show that (mn) is isomorphic to (m) X (n) when m and n are relatively prime. Then we can find out all about (n) by factoring n as a product of powers of primes and using our knowledge of the structure of the groups (p«) . The route we follow is straightforward and comput ational . 10 to write (n) as a product of cyclic groups. Let n = 2«p�1 • • • p�· be the factorization of n into products of primes. 1 Theorem.

We shall also call this extended homomorphism p. If fe R [x] and r e R, then p(f(r)) p(f)(p(r)). In particular, the function assigning to each polynomial with integral = = = 12. The Algebra of PolyJWmials 27 coefficients the polynomial with the same coefficients reduced modulo n is a homomorphism from Z [x] to Zn [x] . When fe Z [x] we write J for its image in Zn [x] ; J is the polynomial whose coefficients are the coefficients of f modulo n. Iff has an integral root r, then clearly r is a root of J, but the converse is false.

Download PDF sample

An Introduction to the Theory of Numbers by G. H. & Wright, E. M. Hardy

by Steven

Rated 4.83 of 5 – based on 35 votes