Prime number

Abstract: From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The Distribution of the Primes.- Riemann's Memoir.- The Functional Equation of the L Function.- Properties of the Gamma Function.- Integral Functions of Order 1.- The Infinite Products for xi(s) and xi(s,Zero-Free Region for zeta(s).- Zero-Free Regions for L(s, chi).- The Number N(T).- The Number N(T, chi).- The explicit Formula for psi(x).- The Prime Number Theorem.- The Explicit Formula for psi(x,chi).- The Prime Number Theorem for Arithmetic Progressions (I).- Siegel's Theorem.- The Prime Number Theorem for Arithmetic Progressions (II).- The Polya-Vinogradov Inequality.- Further Prime Number Sums.

Abstract: 1. The Factorization of Integers.- 1.1 Divisibility.- 1.2 Prime Numbers and Composite Numbers.- 1.3 Prime Numbers.- 1.4 Integral Modulus.- 1.5 The Fundamental Theorem of Arithmetic.- 1.6 The Greatest Common Factor and the Least Common Multiple.- 1.7 The Inclusion-Exclusion Principle.- 1.8 Linear Indeterminate Equations.- 1.9 Perfect Numbers.- 1.10 Mersenne Numbers and Fermat Numbers.- 1.11 The Prime Power in a Factorial.- 1.12 Integral Valued Polynomials.- 1.13 The Factorization of Polynomials.- Notes.- 2. Congruences.- 2.1 Definition.- 2.2 Fundamental Properties of Congruences.- 2.3 Reduced Residue System.- 2.4 The Divisibility of 2p-1-1 by p2.- 2.5 The Function ?(m).- 2.6 Congruences.- 2.7 The Chinese Remainder Theorem.- 2.8 Higher Degree Congruences.- 2.9 Higher Degree Congruences to a Prime Power Modulus.- 2.10 Wolstenholme's Theorem.- 3. Quadratic Residues.- 3.1 Definitions and Euler's Criterion.- 3.2 The Evaluation of Legendre's Symbol.- 3.3 The Law of Quadratic Reciprocity.- 3.4 Practical Methods for the Solutions.- 3.5 The Number of Roots of a Quadratic Congruence.- 3.6 Jacobi's Symbol.- 3.7 Two Terms Congruences.- 3.8 Primitive Roots and Indices.- 3.9 The Structure of a Reduced Residue System.- 4. Properties of Polynomials.- 4.1 The Division of Polynomials.- 4.2 The Unique Factorization Theorem.- 4.3 Congruences.- 4.4 Integer Coefficients Polynomials.- 4.5 Polynomial Congruences with a Prime Modulus.- 4.6 On Several Theorems Concerning Factorizations.- 4.7 Double Moduli Congruences.- 4.8 Generalization of Fermat's Theorem.- 4.9 Irreducible Polynomials mod p.- 4.10 Primitive Roots.- 4.11 Summary.- 5. The Distribution of Prime Numbers.- 5.1 Order of Infinity.- 5.2 The Logarithm Function.- 5.3 Introduction.- 5.4 The Number of Primes is Infinite.- 5.5 Almost All Integers are Composite.- 5.6 Chebyshev's Theorem.- 5.7 Bertrand's Postulate.- 5.8 Estimation of a Sum by an Integral.- 5.9 Consequences of Chebyshev's Theorem.- 5.10 The Number of Prime Factors of n.- 5.11 A Prime Representing Function.- 5.12 On Primes in an Arithmetic Progression.- Notes.- 6. Arithmetic Functions.- 6.1 Examples of Arithmetic Functions.- 6.2 Properties of Multiplicative Functions.- 6.3 The Mobius Inversion Formula.- 6.4 The Mobius Transformation.- 6.5 The Divisor Function.- 6.6 Two Theorems Related to Asymptotic Densities.- 6.7 The Representation of Integers as a Sum of Two Squares.- 6.8 The Methods of Partial Summation and Integration.- 6.9 The Circle Problem.- 6.10 Farey Sequence and Its Applications.- 6.11 Vinogradov's Method of Estimating Sums of Fractional Parts.- 6.12 Application of Vinogradov's Theorem to Lattice Point Problems.- 6.13 ?-results.- 6.14 Dirichlet Series.- 6.15 Lambert Series.- Notes.- 7. Trigonometric Sums and Characters.- 7.1 Representation of Residue Classes.- 7.2 Character Functions.- 7.3 Types of Characters.- 7.4 Character Sums.- 7.5 Gauss Sums.- 7.6 Character Sums and Trigonometric Sums.- 7.7 From Complete Sums to Incomplete Sums.- 7.8 Applications of the Character Sum $$\sum\limits_{x = 1}^p {\left( {\frac{{x^2 + ax + b}}{p}} \right)} $$.- 7.9 The Problem of the Distribution of Primitive Roots.- 7.10 Trigonometric Sums Involving Polynomials.- Notes.- 8. On Several Arithmetic Problems Associated with the Elliptic Modular Function.- 8.1 Introduction.- 8.2 The Partition of Integers.- 8.3 Jacobi's Identity.- 8.4 Methods of Representing Partitions.- 8.5 Graphical Method for Partitions.- 8.6 Estimates for p(n).- 8.7 The Problem of Sums of Squares.- 8.8 Density.- 8.9 A Summary of the Problem of Sums of Squares.- 9. The Prime Number Theorem.- 9.1 Introduction.- 9.2 The Riemann ?-Function.- 9.3 Several Lemmas.- 9.4 A Tauberian Theorem.- 9.5 The Prime Number Theorem.- 9.6 Selberg's Asymptotic Formula.- 9.7 Elementary Proof of the Prime Number Theorem.- 9.8 Dirichlet's Theorem.- Notes.- 10. Continued Fractions and Approximation Methods.- 10.1 Simple Continued Fractions.- 10.2 The Uniqueness of a Continued Fraction Expansion.- 10.3 The Best Approximation.- 10.4 Hurwitz's Theorem.- 10.5 The Equivalence of Real Numbers.- 10.6 Periodic Continued Fractions.- 10.7 Legendre's Criterion.- 10.8 Quadradic Indeterminate Equations.- 10.9 Pell's Equation.- 10.10 Chebyshev's Theorem and Khintchin's Theorem.- 10.11 Uniform Distributions and the Uniform Distribution of n? (mod 1).- 10.12 Criteria for Uniform Distributions.- 11. Indeterminate Equations.- 11.1 Introduction.- 11.2 Linear Indeterminate Equations.- 11.3 Quadratic Indeterminate Equations.- 11.4 The Solution to ax2 + bxy + cy2=k.- 11.5 Method of Solution.- 11.6 Generalization of Soon Go's Theorem.- 11.7 Fermat's Conjecture.- 11.8 Markoff's Equation.- 11.9 The Equation x3 + y3 + z3 + ?3=0.- 11.10 Rational Points on a Cubic Surface.- Notes.- 12. Binary Quadratic Forms.- 12.1 The Partitioning of Binary Quadratic Forms into Classes.- 12.2 The Finiteness of the Number of Classes.- 12.3 Kronecker's Symbol.- 12.4 The Number of Representations of an Integer by a Form.- 12.5 The Equivalence of Formsmod q.- 12.6 The Character System for a Quadratic Form and the Genus.- 12.7 The Convergence of the Series K(d).- 12.8 The Number of Lattice Points Inside a Hyperbola and an Ellipse.- 12.9 The Limiting Average.- 12.10 The Class Number: An Analytic Expression.- 12.11 The Fundamental Discriminants.- 12.12 The Class Number Formula.- 12.13 The Least Solution to Pell's Equation.- 12.14 Several Lemmas.- 12.15 Siegel's Theorem.- Notes.- 13. Unimodular Transformations.- 13.1 The Complex Plane.- 13.2 Properties of the Bilinear Transformation.- 13.3 Geometric Properties of the Bilinear Transformation.- 13.4 Real Transformations.- 13.5 Unimodular Transformations.- 13.6 The Fundamental Region.- 13.7 The Net of the Fundamental Region.- 13.8 The Structure of the Modular Group.- 13.9 Positive Definite Quadratic Forms.- 13.10 Indefinite Quadratic Forms.- 13.11 The Least Value of an Indefinite Quadratic Form.- 14. Integer Matrices and Their Applications.- 14.1 Introduction.- 14.2 The Product of Matrices.- 14.3 The Number of Generators for Modular Matrices.- 14.4 Left Association.- 14.5 Invariant Factors and Elementary Divisors.- 14.6 Applications.- 14.7 Matrix Factorizations and Standard Prime Matrices.- 14.8 The Greatest Common Factor and the Least Common Multiple.- 14.9 Linear Modules.- 15. p-adic Numbers.- 15.1 Introduction.- 15.2 The Definition of a Valuation.- 15.3 The Partitioning of Valuations into Classes.- 15.4 Archimedian Valuations.- 15.5 Non-Archimedian Valuations.- 15.6 The ?-Extension of the Rationals.- 15.7 The Completeness of the Extension.- 15.8 The Representation of p-adic Numbers.- 15.9 Application.- 16. Introduction to Algebraic Number Theory.- 16.1 Algebraic Numbers.- 16.2 Algebraic Number Fields.- 16.3 Basis.- 16.4 Integral Basis.- 16.5 Divisibility.- 16.6 Ideals.- 16.7 Unique Factorization Theorem for Ideals.- 16.8 Basis for Ideals.- 16.9 Congruent Relations.- 16.10 Prime Ideals.- 16.11 Units.- 16.12 Ideal Classes.- 16.13 Quadratic Fields and Quadratic Forms.- 16.14 Genus.- 16.15 Euclidean Fields and Simple Fields.- 16.16 Lucas's Criterion for the Determination of Mersenne Primes.- 16.17 Indeterminate Equations.- 16.18 Tables.- Notes.- 17. Algebraic Numbers and Transcendental Numbers.- 17.1 The Existence of Transcendental Numbers.- 17.2 Liouville's Theorem and Examples of Transcendental Numbers.- 17.3 Roth's Theorem on Rational Approximations to Algebraic Numbers.- 17.4 Application of Roth's Theorem.- 17.5 Application of Thue's Theorem.- 17.6 The Transcendence of e.- 17.7 The Transcendence of ?.- 17.8 Hilbert's Seventh Problem.- 17.9 Gelfond's Proof.- Notes.- 18. Waring's Problem and the Problem of Prouhet and Tarry.- 18.1 Introduction.- 18.2 Lower Bounds for g(k) and G(k).- 18.3 Cauchy's Theorem.- 18.4 Elementary Methods.- 18.5 The Easier Problem of Positive and Negative Signs.- 18.6 Equal Power Sums Problem.- 18.7 The Problem of Prouhet and Tarry.- 18.8 Continuation.- 19. Schnirelmann Density.- 19.1 The Definition of Density and its History.- 19.2 The Sum of Sets and its Density.- 19.3 The Goldbach-Schnirelmann Theorem.- 19.4 Selberg's Inequality.- 19.5 The Proof of the Goldbach-Schnirelmann Theorem.- 19.6 The Waring-Hiibert Theorem.- 19.7 The Proof of the Waring-Hiibert Theorem.- Notes.- 20. The Geometry of Numbers.- 20.1 The Two Dimensional Situation.- 20.2 The Fundamental Theorem of Minkowski.- 20.3 Linear Forms.- 20.4 Positive Definite Quadratic Forms.- 20.5 Products of Linear Forms.- 20.6 Method of Simultaneous Approximations.- 20.7 Minkowski's Inequality.- 20.8 The Average Value of the Product of Linear Forms.- 20.9 Tchebotaref's Theorem.- 20.10 Applications to Algebraic Number Theory.- 20.11 The Least Value for |?|.

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: In this table, g is the genus of Xo(N), and v the number of noncuspidal rational points of Xo(N) (which is, in effect, the number of rational N-isogenies classified up to "twist"). For an excellent readable account of isogenies and their related diophantine problems, see Ogg's [25, 26]. The first column of the table corresponds to the genus 0 cases; for each of these values of N rational parametrizat ions of Xo(N) are known [10]. For each integer N, and each order R ~ Q(1/-ZN) such that R contains ~ and has class number one, there is a Q-rat ional N-isogeny. This accounts for one noncuspidal rational point on Xo(N ) for N = 11, 19, 27, 43, 67, 163 and for the two noncuspidal rational points on Xo(14 ). For a discussion of the cases: N = l l , 15, 17, 21 see ([43], pp.78-80) and for the peculiar N = 37, see ([22], w 5). The object of this paper is to show that when N is a prime number there are no Q-rat ional N-isogenies beyond those exhibited in the above table. To prove this (in the light of known results concerning Xo(N)(Q) for the twelve prime numbers N appearing in the table [19]) it suffices to show: