Marshall D.C., Odell E., Starbird M. Number Theory through Inquiry

Washington, D.C.: Mathematical Association of America, 2007. — 150 p. — (MAA textbooks). — ISBN: 978-0-88385-983-4.This innovative textbook leads students on a carefully guided discovery of introductory number theory. The book has two equally significant goals. The first is to help students develop mathematical thinking skills, particularly theorem-proving skills. The other goal is to help students understand some of the wonderfully rich ideas in the mathematical study of numbers. This book is appropriate for a proof transitions course, for independent study, or for a course designed as an introduction to abstract mathematics. It is designed to be used with an instructional technique variously called guided discovery or Modified Moore Method or Inquiry Based Learning (IBL). Instructors' materials explain the instructional method, which gives students a totally different experience compared to a standard lecture course. Students develop an attitude of personal reliance and a sense that they can think effectively about difficult problems; goals that are fundamental to the educational enterprise within and beyond mathematics.Number Theory and Mathematical Thinking
Note on the approach and organization
Methods of thoughtDivide and Conquer
Divisibility in the Natural Numbers
Definitions and examples
Divisibility and congruence
The Division Algorithm
Greatest common divisors and linear Diophantine equations
Linear Equations Through the AgesPrime Time
The Prime Numbers
Fundamental Theorem of Arithmetic
Applications of the Fundamental Theorem of Arithmetic
The infinitude of primes
Primes of special form
The distribution of primes
From Antiquity to the InternetA Modular World
Thinking Cyclically
Powers and polynomials modulo n
Linear congruences
Systems of linear congruences: the Chinese Remainder Theorem
A Prince and a MasterFermat’s Little Theorem and Euler’s Theorem
Abstracting the Ordinary
Orders of an integer modulo n
Fermat’s Little Theorem
An alternative route to Fermat’s Little Theorem
Euler’s Theorem and Wilson’s Theorem
Fermat, Wilson and...Leibniz?Public Key Cryptography
Public Key Codes and RSA
Public key codes
Overview of RSA
Let’s decryptPolynomial Congruences and Primitive Roots
Higher Order Congruences
Lagrange’s Theorem
Primitive roots
Euler’s φ-function and sums of divisors
Euler’s φ-function is multiplicative
Roots modulo a number
Sophie Germain is Germane, Part IThe Golden Rule: Quadratic Reciprocity
Quadratic Congruences
Quadratic residues
Gauss’ Lemma and quadratic reciprocity
Sophie Germain is germane, Part IIPythagorean Triples, Sums of Squares, and Fermat’s Last Theorem
Congruences to Equations
Pythagorean triples
Sums of squares
Pythagorean triples revisited
Fermat’s Last Theorem
Who’s Represented?
Sums of squares
Sums of cubes, taxicabs, and Fermat’s Last TheoremRationals Close to Irrationals and the Pell Equation
Diophantine Approximation and Pell Equations
A plunge into rational approximation
Out with the trivial
New solutions from old
Securing the elusive solution
The structure of the solutions to the Pell equations
Bovine MathThe Search for Primes
Primality Testing
Is it prime?
Fermat’s Little Theorem and probable primes
AKS primality
Record PrimesAppendix. Mathematical Induction: The Domino Effect
The Infinitude Of Facts
Gauss’ formula
Another formula
On your own
Strong induction
On your ownIndex
About the Authors