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by Jeanine Meyer, Marty Lewinter
Elementary Number Theory with Programming
COVER
TITLE PAGE
PREFACE
WORDS
NOTATION IN MATHEMATICAL WRITING AND IN PROGRAMMING
1 SPECIAL NUMBERS: TRIANGULAR, OBLONG, PERFECT, DEFICIENT, AND ABUNDANT
TRIANGULAR NUMBERS
OBLONG NUMBERS AND SQUARES
DEFICIENT, ABUNDANT, AND PERFECT NUMBERS
EXERCISES
2 FIBONACCI SEQUENCE, PRIMES, AND THE PELL EQUATION
PRIME NUMBERS AND PROOF BY CONTRADICTION
PROOF BY CONSTRUCTION
SUMS OF TWO SQUARES
BUILDING A PROOF ON PRIOR ASSERTIONS
SIGMA NOTATION
SOME SUMS
FINDING ARITHMETIC FUNCTIONS
FIBONACCI NUMBERS
AN INFINITE PRODUCT
THE PELL EQUATION
GOLDBACH’S CONJECTURE
EXERCISES
3 PASCAL’S TRIANGLE
FACTORIALS
THE COMBINATORIAL NUMBERS n CHOOSE k
PASCAL’S TRIANGLE
BINOMIAL COEFFICIENTS
EXERCISES
4 DIVISORS AND PRIME DECOMPOSITION
DIVISORS
GREATEST COMMON DIVISOR
DIOPHANTINE EQUATIONS
LEAST COMMON MULTIPLE
PRIME DECOMPOSITION
SEMIPRIME NUMBERS
WHEN IS A NUMBER AN mTH POWER?
TWIN PRIMES
FERMAT PRIMES
ODD PRIMES ARE DIFFERENCES OF SQUARES
WHEN IS n A LINEAR COMBINATION OF a AND b?
PRIME DECOMPOSITION OF n!
NO NONCONSTANT POLYNOMIAL WITH INTEGER COEFFICIENTS ASSUMES ONLY PRIME VALUES
EXERCISES
5 MODULAR ARITHMETIC
CONGRUENCE CLASSES MOD k
LAWS OF MODULAR ARITHMETIC
MODULAR EQUATIONS
FERMAT’S LITTLE THEOREM
FERMAT’S LITTLE THEOREM
MULTIPLICATIVE INVERSES
WILSON’S THEOREM
WILSON’S THEOREM
WILSON’S THEOREM (2ND VERSION)
SQUARES AND QUADRATIC RESIDUES
LAGRANGE’S THEOREM
LAGRANGE’S THEOREM
REDUCED PYTHAGOREAN TRIPLES
CHINESE REMAINDER THEOREM
CHINESE REMAINDER THEOREM
EXERCISES
6 NUMBER THEORETIC FUNCTIONS
THE TAU FUNCTION
THE SIGMA FUNCTION
MULTIPLICATIVE FUNCTIONS
PERFECT NUMBERS REVISITED
MERSENNE PRIMES
F(n) = ∑) = ∑f(d) WHERE d IS A DIVISOR OF n
THE MÖBIUS FUNCTION
THE RIEMANN ZETA FUNCTION
EXERCISES
7 THE EULER PHI FUNCTION
THE PHI FUNCTION
EULER’S GENERALIZATION OF FERMAT’S LITTLE THEOREM
PHI OF A PRODUCT OF m AND n WHEN gcd(m,n) > 1) > 1
THE ORDER OF a (mod n)
PRIMITIVE ROOTS
THE INDEX OF m (mod p) RELATIVE TO a
TO BE OR NOT TO BE A QUADRATIC RESIDUE
THE LEGENDRE SYMBOL
QUADRATIC RECIPROCITY
LAW OF QUADRATIC RECIPROCITY
WHEN DOES x2 = a (mod n) HAVE A SOLUTION?
EXERCISES
8 SUMS AND PARTITIONS
AN nTH POWER IS THE SUM OF TWO SQUARES
SOLUTIONS TO THE DIOPHANTINE EQUATION a2 + b2 + c2 = d2
ROW SUMS OF A TRIANGULAR ARRAY OF CONSECUTIVE ODD NUMBERS
PARTITIONS
WHEN IS A NUMBER THE SUM OF TWO SQUARES?
SUMS OF FOUR OR FEWER SQUARES
EXERCISES
9 CRYPTOGRAPHY
INTRODUCTION AND HISTORY
PUBLIC-KEY CRYPTOGRAPHY
FACTORING LARGE NUMBERS
THE KNAPSACK PROBLEM
SUPERINCREASING SEQUENCES
EXERCISES
ANSWERS OR HINTS TO SELECTED EXERCISES
CHAPTER 1
CHAPTER 2
CHAPTER 3
CHAPTER 4
CHAPTER 5
CHAPTER 6
CHAPTER 7
CHAPTER 8
CHAPTER 9
INDEX
END USER LICENSE AGREEMENT
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Elementary Number Theory with Programming
ELEMENTARY NUMBER THEORY WITH PROGRAMMING
MARTY LEWINTER
JEANINE MEYER
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