Elementary Number Theory with Programming (eBook)

Marty Lewinter, PhD, is Professor Emeritus of Mathematics at the State University of New York, Purchase College. The author of three books and more than 80 articles, he is Executive Director of Mathematics at American Digital University Services. Jeanine Meyer, PhD, is Professor of Mathematics/Computer Science at the State University of New York, Purchase College. She is the author of six books as well as numerous journal articles.

Chapter I: Pythagoras: "Everything is number"

This first chapter presents several definitions for classes ofnumber, including triangular and perfect. The programs include onefor factoring numbers and one to test a conjecture up to a fixedlimit

Chapter II: Proof

The second chapter focuses on primes as well as approaches forsolving Pell equations. The programs include examples that countsteps to compare two different approaches

Chapter III: Pascal's Triangle

The third chapter focuses on factorial. It also presentsPascal's Triangle. The programs include examples thatgenerate factorial using iteration and using recursion and thusdemonstrate and compare important techniques in programming

Chapter IV: Divisors and Primes

The fourth chapter returns to factoring, demonstrating thealgorithm for producing the greatest common divisor of two numbers.The programs include one that uses the algorithm to produce the GCDof a pair of numbers and a program to produce the primedecomposition of a number

Chapter V: Modular Arithmetic

The fifth chapter presents mod equations. One program checks ifa mod equation is true and another determines the solvability of amod equation and then solves an equation that is solvable by abrute force approach

Chapter VI: Number Theoretic Functions

The sixth chapter is again on factoring and also the Taufunction. The programs include two distinct approaches tocalculating the Tau function

Chapter VII: Euler's Phi Function

The seventh chapter presents the Euler Phi function and otherfunctions. The programs demonstrate two approaches to calculatingthe Phi function

Chapter VIII: Sums and Partitions

The eighth chapter presents partitions, including binarypartitions. The exposition explains the central role of binaryrepresentation in computing and the programs produce the binarypartition using a built-in function and also using the mathematicalapproach explained in the chapter

Chapter IX: Cryptography

The ninth chapter presents codes from very old to modern day.The programs include different ways to generate counts of lettersand also Fermat factoring

Answers or hints to selected exercises

Sample programs at the end of each chapter. You canaccess working examples of the sample programs at thewebsite: http://faculty.purchase.edu/jeanine.meyer/numbertheory/

"It consists of nine chapters, all including the corresponding programs along with their mathematical content. The mathematical structure is also interesting and well-formed starting from special numbers, primes and Pell equation, to Pascal's triangle, prime decomposition and modular arithmetic and finishing with number-theoretic functions, the Euler Phi-function, sums and partitions and the classical application to cryptography. It is also remarkable that the main scope of the programs is defined before their use from the reader, providing him the best orientation for his study." (Zentralblatt MATH 2016)