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Full Description
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.
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
