Quadratic integer

Abstract: P. What is Number Theory? 1. The Integers. Numbers and Sequences. Sums and Products. Mathematical Induction. The Fibonacci Numbers. 2. Integer Representations and Operations. Representations of Integers. Computer Operations with Integers. Complexity of Integer Operations. 3. Primes and Greatest Common Divisors. Prime Numbers. The Distribution of Primes. Greatest Common Divisors. The Euclidean Algorithm. The Fundemental Theorem of Arithmetic. Factorization Methods and Fermat Numbers. Linear Diophantine Equations. 4. Congruences. Introduction to Congruences. Linear Congrences. The Chinese Remainder Theorem. Solving Polynomial Congruences. Systems of Linear Congruences. Factoring Using the Pollard Rho Method. 5. Applications of Congruences. Divisibility Tests. The perpetual Calendar. Round Robin Tournaments. Hashing Functions. Check Digits. 6. Some Special Congruences. Wilson's Theorem and Fermat's Little Theorem. Pseudoprimes. Euler's Theorem. 7. Multiplicative Functions. The Euler Phi-Function. The Sum and Number of Divisors. Perfect Numbers and Mersenne Primes. Mobius Inversion. 8. Cryptology. Character Ciphers. Block and Stream Ciphers. Exponentiation Ciphers. Knapsack Ciphers. Cryptographic Protocols and Applications. 9. Primitive Roots. The Order of an Integer and Primitive Roots. Primitive Roots for Primes. The Existence of Primitive Roots. Index Arithmetic. Primality Tests Using Orders of Integers and Primitive Roots. Universal Exponents. 10. Applications of Primitive Roots and the Order of an Integer. Pseudorandom Numbers. The EIGamal Cryptosystem. An Application to the Splicing of Telephone Cables. 11. Quadratic Residues. Quadratic Residues and nonresidues. The Law of Quadratic Reciprocity. The Jacobi Symbol. Euler Pseudoprimes. Zero-Knowledge Proofs. 12. Decimal Fractions and Continued. Decimal Fractions. Finite Continued Fractions. Infinite Continued Fractions. Periodic Continued Fractions. Factoring Using Continued Fractions. 13. Some Nonlinear Diophantine Equations. Pythagorean Triples. Fermat's Last Theorem. Sums of Squares. Pell's Equation. 14. The Gaussian Integers. Gaussian Primes. Unique Factorization of Gaussian Integers. Gaussian Integers and Sums of Squares.

Abstract: For each integer r > 1 we will define an arithmetic function H(r, N) which for r = 1 is the class number of (not necessarily primitive) quadratic forms of discriminant N and for r > 1 is essentially the value of ~K(r) where K = ~ ( ] / ~ ) . For r > 2 the function ~N>_0 H(r, N)e 2~i~ is a modular form of weight r + 1/2 on Fo(4 ). This implies numerous identities involving H(r, N). The analogous formulas for r = 1 (which do not follow from the methods of this paper) are classical "class number relations" of Kronecker, Hurwitz and others, as well as certain generalizations coming from the Selberg-Eichler trace formula and from recent work of Hirzebruch-Zagier. One of the tools used is of independent interest: given two modular f o r m s f a n d 9, there are certain bilinear expressions in the derivatives o f f and 9 which are again modular forms.

Abstract: (1.1) a1x1 + . . .+ akxk = b with x1, . . . , xk in a prescribed set of integers. We saw that the vanishing of the constant term b and the sum of coefficients s = a1 + . . . + ak had a strong effect on the behaviour of equation (1.1). The condition b = 0 is equivalent to homogeneity or multiplication invariance (if x1, . . . , xk is a solution, so is tx1, . . . , txk), while s = 0 means translation invariance (if x1, . . . , xk is a solution, so is x1 + t, . . . , xk+ t). We called equations with b = s = 0 invariant , and those with b 6= 0 or s 6= 0 noninvariant . In Part I of the paper we studied invariant equations; now we treat noninvariant ones. We recall the principal notations.