Sums of squares of integers

Introduction Prerequisites Outline of Chapters 2 - 8 Elementary Methods Introduction Some Lemmas Two Fundamental Identities Euler's Recurrence for Sigma(n)More Identities Sums of Two Squares Sums of Four Squares Still More Identities Sums of Three Squares An Alternate Method Sums of Polygonal Numbers Exercises Bernoulli Numbers Overview Definition of the Bernoulli Numbers The Euler-MacLaurin Sum Formula The Riemann Zeta Function Signs of Bernoulli Numbers Alternate The von Staudt-Clausen Theorem Congruences of Voronoi and Kummer Irregular Primes Fractional Parts of Bernoulli Numbers Exercises Examples of Modular Forms Introduction An Example of Jacobi and Smith An Example of Ramanujan and Mordell An Example of Wilton: t (n) Modulo 23 An Example of Hamburger Exercises Hecke's Theory of Modular FormsIntroduction Modular Group ? and its Subgroup ? 0 (N) Fundamental Domains For ? and ? 0 (N) Integral Modular Forms Modular Forms of Type Mk(? 0(N);chi) and Euler-Poincare series Hecke Operators Dirichlet Series and Their Functional Equation The Petersson Inner Product The Method of Poincare Series Fourier Coefficients of Poincare Series A Classical Bound for the Ramanujan t functionThe Eichler-Selberg Trace Formula l-adic Representations and the Ramanujan Conjecture Exercises Representation of Numbers as Sums of Squares Introduction The Circle Method and Poincare Series Explicit Formulas for the Singular Series The Singular Series Exact Formulas for the Number of Representations Examples: Quadratic Forms and Sums of Squares Liouville's Methods and Elliptic Modular Forms Exercises Arithmetic Progressions Introduction Van der Waerden's Theorem Roth's Theorem t 3 = 0 Szemeredi's Proof of Roth's Theorem Bipartite Graphs Configurations More Definitions The Choice of tm Well-Saturated K-tuples Szemeredi's Theorem Arithmetic Progressions of Squares Exercises Applications Factoring Integers Computing Sums of Two Squares Computing Sums of Three Squares Computing Sums of Four Squares Computing rs(n) Resonant Cavities Diamond Cutting Cryptanalysis of a Signature Scheme Exercises References Index.