Book:J. Hunter/Number Theory

Subject Matter

 * Number Theory

Contents

 * Preface


 * $\text{I}$ NUMBER SYSTEMS AND ALGEBRAIC STRUCTURES
 * 1. Introduction
 * 2. The positive integers
 * 3. Equivalence relations
 * 4. The set of all integers
 * 5. The rational numbers
 * 6. Algebraic structures
 * Examples


 * $\text{II}$ DIVISION AND FACTORISATION PROPERTIES
 * 7. Division identities for the integers
 * 8. Representation in the scale of $g$
 * 9. Least common multiple, greatest common divisor, Euclidean algorithm
 * 10. Prime numbers, unique factorisation theorem
 * 11. Relatively prime numbers, Euler's function $\phi$
 * 12. Multiplicative arithmetical functions, the Möbius function $\mu$, the inversion formula
 * Examples


 * $\text{III}$ CONGRUENCES
 * 13. Congruence notation, operations on congruences
 * 14. Residue sets (mod $m$)
 * 15. Euler's Theorem, order of $a$ (mod $m$)
 * 16. Linear congruences
 * 17. The ring of congruence classes (mod $m$)
 * 18. Algebraic interpretation of Theorems $22$, $23$ and $24$
 * Examples


 * $\text{IV}$ ALGEBRAIC CONGRUENCES AND PRIMITIVE ROOTS
 * 19. Algebraic congruences
 * 20. Algebraic congruences (mod $p$)
 * 21. Algebraic congruences with composite modulus
 * 22. Primitive roots
 * 23. Indices
 * Examples


 * $\text{V}$ QUADRATIC RESIDUES
 * 24. $n$-th power residues
 * 25. The Legendre symbol $\paren {a / p}$
 * 26. The law of quadratic reciprocity
 * 27. The Jacobi symbol $\paren {a / b}$
 * Examples


 * $\text{VI}$ REPRESENTATION OF INTEGERS BY BINARY QUADRATIC FORMS
 * 28. Definitions and notation
 * 29. Unimodular matrices and transformations
 * 30. Equivalence classes of binary quadratic forms
 * 31. Binary quadratic forms of given discriminant $d$
 * 32. Representation of integers by binary quadratic forms
 * 33. Representation of an integer as a sum of two squares
 * Examples


 * $\text{VII}$ SOME DIOPHANTINE EQUATIONS
 * 34. Diophantine equations
 * 35. Linear diophantine equations
 * 36. The equation $x^2 + y^2 = z^2$, and related equations
 * 37. Fermat's Last Theorem, the equation $x^4 + y^4 = z^2$
 * Examples


 * Index



Source work progress
* : Chapter $\text {I}$: Number Systems and Algebraic Structures: $2$. The positive integers
 * $1$-based exposition of Peano structure to be embarked upon.