CoverPrefaceContents1 Natural numbers and integers1.1 Natural numbers1.2 Induction1.3 Integers1.4 Division with remainder1.5 Binary notation1.6 Diophantine equations1.7 The Diophantus chord method1.8 Gaussian integers1.9 Discussion

2 The Euclidean algorithm2.1 The gcd by subtraction2.2 The gcd by division with remainder2.3 Linear representation of the gcd2.4 Primes and factorization2.5 Consequences of unique prime factorization2.6 Linear Diophantine equations2.7 *The vector Euclidean algorithm2.8 *The map of relatively prime pairs2.9 Discussion

3 Congruence arithmetic3.1 Congruence mod n3.2 Congruence classes and their arithmetic3.3 Inverses mod p3.4 Fermat's little theorem3.5 Congruence theorems of Wilson and Lagrange3.6 Inverses mod k3.7 Quadratic Diophantine equations3.8 *Primitive roots3.9 *Existence of primitive roots3.10 Discussion

4 The RSA cryptosystem4.1 Trapdoor functions4.2 Ingredients of RSA4.3 Exponentiation mod n4.4 RSA encryption and decryption4.5 Digital signatures4.6 Other computational issues4.7 Discussion

5 The Pell equation5.1 Side and diagonal numbers5.2 The equation x2y = 15.3 The group of solutions5.4 The general Pell equation and Z[sqrt(n)]5.5 The pigeonhole argument5.6 *Quadratic forms5.7 *The map of primitive vectors5.8 *Periodicity in the map of xny5.9 Discussion

6 The Gaussian integers6.1 Z[i] and its norm6.2 Divisibility and primes in Z[i] and Z6.3 Conjugates6.4 Division in Z[i]6.5 Fermat's two square theorem6.6 Pythagorean triples6.7 *Primes of the form 4n+16.8 Discussion

7 Quadratic integers7.1 The equation y = x+27.2 The division property in Z[sqrt(2)]7.3 The gcd in Z[sqrt(-2)]7.4 Z[sqrt(-3)] and Z[zeta]7.5 *Rational solutions of x+y = z+w7.6 "The prime sqrt(-3) in Z[zeta]7.7 *Fermat's last theorem for n = 37.8 Discussion

8 The four square theorem8.1 Real matrices and C8.2 Complex matrices and H8.3 The quaternion units8.4 Z[i,j,k]8.5 The Hurwitz integers8.6 Conjugates8.7 A prime divisor property8.8 Proof of the four square theorem8.9 Discussion

9 Quadratic reciprocity9.1 Primes x+y, x2+2y, and x+3y9.2 Statement of quadratic reciprocity9.3 Euler's criterion9.4 The value of (2/q)9.5 The story so far9.6 The Chinese remainder theorem9.7 The full Chinese remainder theorem9.8 Proof of quadratic reciprocity9.9 Discussion

10 Rings10.1 The ring axioms10.2 Rings and fields10.3 Algebraic integers10.4 Quadratic fields and their integers10.5 Norm and units of quadratic fields10.6 Discussion

11 Ideals11.1 Ideals and the gcd11.2 Ideals and divisibility in ?11.3 Principal ideal domains11.4 A nonprincipal ideal of Z[sqrt(-3)]11.5 A nonprincipal ideal of Z[sqrt(5)]11.6 Ideals of imaginary quadratic fields as lattices11.7 Products and prime ideals11.8 Ideal prime factorization11.9 Discussion

12 Prime ideals12.1 Ideals and congruence12.2 Prime and maximal ideals12.3 Prime ideals of imaginary quadratic fields12.4 Conjugate ideals12.5 Divisibility and containment12.6 Factorization of ideals12.7 Ideal classes12.8 Primes of the form x+5y12.9 Discussion

BibliographyIndex