Number Theory Revealed: A Masterclass

ISBN-13: 9781470463700 / Angielski

Andrew Granville

Opis

Number Theory Revealed: A Masterclass acquaints enthusiastic students with the ""Queen of Mathematics''. The text offers a fresh take on congruences, power residues, quadratic residues, primes, and Diophantine equations and presents hot topics like cryptography, factoring, and primality testing. Students are also introduced to beautiful enlightening questions like the structure of Pascal's triangle mod $p$ and modern twists on traditional questions like the values represented by binary quadratic forms, the anatomy of integers, and elliptic curves. This Masterclass edition contains many additional chapters and appendices not found in Number Theory Revealed: An Introduction, highlighting beautiful developments and inspiring other subjects in mathematics (like algebra). This allows instructors to tailor a course suited to their own (and their students') interests. There are new yet accessible topics like the curvature of circles in a tiling of a circle by circles, the latest discoveries on gaps between primes, a new proof of Mordell's Theorem for congruent elliptic curves, and a discussion of the $abc$-conjecture including its proof for polynomials.

Spis treści

• Preliminary chapter on induction
• The Eulidean algorithm
• Congruences
• The basic algebra of number theory
• Multiplicative functions
• The distribution of prime numbers
• Diophantine problems
• Power residues
• Quadratic residues
• Quadratic equations
• Square roots and factoring
• Rational approximations to real numbers
• Binary quadratic forms
• The anatomy of integers
• Counting integral and rational points on curves, modulo $p$
• Combinatorial number theory
• The $p$-adic numbers
• Rational points on elliptic curve
• Hints for exercises
• Recommended further reading
• Index