(German for Lectures on Number Theory) is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Others were written by Leopold Kronecker, Edmund Landau, and Helmut Hasse. They all cover elementary number theory, Dirichlet's theorem, quadratic fields and forms, and sometimes more advanced topics. Dirichlet and Dedekind's book Based on Dirichlet's number theory course at the University of Göttingen, the were edited by Dedekind and published after Lejeune Dirichlet's death. Dedekind added several appendices to the , in which he collected further results of Lejeune Dirichlet's and also developed his own original mathematical ideas. Scope The cover topics in elementary number theory, algebraic number theory and analytic number theory, including modular arithmetic, quadratic congruences, quadratic reciprocity and binary quadratic forms. Contents The contents of Professor John Stillwell's 1999 translation of the are as follows Chapter 1. On the divisibility of numbers Chapter 2. On the congruence of numbers Chapter 3. On quadratic residues Chapter 4. On quadratic forms Chapter 5. Determination of the class number of binary quadratic forms Supplement I. Some theorems from Gauss's theory of circle division Supplement II. On the limiting value of an infinite series Supplement III. A geometric theorem Supplement IV. Genera of quadratic forms Supplement V. Power residues for composite moduli Supplement VI. Primes in arithmetic progressions Supplement VII. Some theorems from the theory of circle division Supplement VIII. On the Pell equation Supplement IX. Convergence and continuity of some infinite series This translation does not include Dedekind's Supplements X and XI in which he begins to develop the theory of ideals. The German titles of supplements X and XI are: Supplement X: Über die Composition der binären quadratische Formen (On the composition of binary quadratic forms) Supplement XI: Über die Theorie der ganzen algebraischen Zahlen (On the theory of algebraic integers) Chapters 1 to 4 cover similar ground to Gauss' , and Dedekind added footnotes which specifically cross-reference the relevant sections of the . These chapters can be thought of as a summary of existing knowledge, although Dirichlet simplifies Gauss' presentation, and introduces his own proofs in some places. Chapter 5 contains Dirichlet's derivation of the class number formula for real and imaginary quadratic fields. Although other mathematicians had conjectured similar formulae, Dirichlet gave the first rigorous proof. Supplement VI contains Dirichlet's proof that an arithmetic