Table of Contents

Numbers The problem $Asquare + B = square$ Congruences Double congruences and the Euclidean algorithm The augmented Euclidean algorithm Simultaneous congruences The fundamental theorem of arithmetic Exponentiation and orders Euler's $phi$-function Finding the order of $abmod c$ Primality testing The RSA cipher system Primitive roots $bmodp$ Polynomials Tables of indices $bmod p$ Brahmagupta's formula and hypernumbers Modules of hypernumbers A canonical form for modules of hypernumbers Solution of $Asquare + B = square$ Proof of the theorem of Chapter 19 Euler's remarkable discovery Stable modules Equivalence of modules Signatures of equivalence classes The main theorem Which modules become principal when squared? The possible signatures for certain values of $A$ The law of quadratic reciprocity Proof of the Main Theorem The theory of binary quadratic forms Composition of binary quadratic forms Cycles of stable modules Answers to exercises Bibliography Index.