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Full Description
Number Theory Revealed: A Masterclass presents 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, Fermat's Last Theorem for polynomials, and modern twists on traditional questions.
This Masterclass edition contains many additional chapters and appendices not found in Number Theory Revealed: An Introduction. It is ideal for instructors who wish to tailor a class to their own interests and gives well-prepared students further opportunities to challenge themselves and push beyond core number theory concepts, serving as a springboard to many current themes in mathematics. Additional topics in A Masterclass include the curvature of circles in a tiling of a circle by circles, the latest discoveries on gaps between primes, magic squares of primes, a new proof of Mordell's Theorem for congruent elliptic curves, as well as links with algebra, analysis, cryptography, and dynamics.
Contents
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.
