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**Elementary number theory. Translated from the German by J. E. Goodman. With added exercises by P. T. Bateman and Eugene Kohlbecker.**
*(English)*
Zbl 0079.06201

New York: Chelsea Publishing Co. 250 p. $ 4.95 (1958).

This is a translation of the first part of the first book of Landau’s famous “Elementare Zahlentheorie” (Hirzel 1927; JFM 53.0123.17) with added exercises by Paul T. Bateman and Eugene E. Kohlbecker. The remaining parts have been published by Chelsea Publishing Co. in 1969.

Table of Contents:

Part One. Foundations of Number Theory

The greatest common divisor of two numbers. Prime numbers and factorization into prime factors. The greatest common divisor of several numbers. Number-theoretic functions. Congruences. Quadratic residues. Pell’s equation.

Part Two. Brun’s Theorem and Dirichlet’s Theorem

Introduction. Some elementary inequalities of prime number theory. Brun’s theorem on prime pairs. Dirichlet’s theorem on the prime numbers in an arithmetic progression; Further theorems on congruences; Characters; \(L\)-series; Dirichlet’s proof.

Part Three. Decomposition into Two, Three, and Four Squares

Introduction. Farey fractions. Decomposition into two squares. Decomposition into four squares; Introduction; Lagrange’s theorem; Determination of the number of solutions. Decomposition into three squares; Equivalence of quadratic forms; A necessary condition for decomposability into three squares; The necessary condition is sufficient.

Part Four. The Class Number of Binary Quadratic Forms

Introduction. Factorable and unfactorable forms. Classes of forms. The finiteness of the class number. Primary representations by forms. The representation of \(h(d)\) in terms of \(K(d)\). Gaussian sums; Appendix; Introduction; Kronecker’s proof; Schur’s proof; Mertens’ proof. Reduction to fundamental discriminants. The determination of \(K(d)\) for fundamental discriminants. Final formulas for the class number.

Appendix. Exercises.

Table of Contents:

Part One. Foundations of Number Theory

The greatest common divisor of two numbers. Prime numbers and factorization into prime factors. The greatest common divisor of several numbers. Number-theoretic functions. Congruences. Quadratic residues. Pell’s equation.

Part Two. Brun’s Theorem and Dirichlet’s Theorem

Introduction. Some elementary inequalities of prime number theory. Brun’s theorem on prime pairs. Dirichlet’s theorem on the prime numbers in an arithmetic progression; Further theorems on congruences; Characters; \(L\)-series; Dirichlet’s proof.

Part Three. Decomposition into Two, Three, and Four Squares

Introduction. Farey fractions. Decomposition into two squares. Decomposition into four squares; Introduction; Lagrange’s theorem; Determination of the number of solutions. Decomposition into three squares; Equivalence of quadratic forms; A necessary condition for decomposability into three squares; The necessary condition is sufficient.

Part Four. The Class Number of Binary Quadratic Forms

Introduction. Factorable and unfactorable forms. Classes of forms. The finiteness of the class number. Primary representations by forms. The representation of \(h(d)\) in terms of \(K(d)\). Gaussian sums; Appendix; Introduction; Kronecker’s proof; Schur’s proof; Mertens’ proof. Reduction to fundamental discriminants. The determination of \(K(d)\) for fundamental discriminants. Final formulas for the class number.

Appendix. Exercises.

### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

01A75 | Collected or selected works; reprintings or translations of classics |

11Axx | Elementary number theory |

11Nxx | Multiplicative number theory |

11E12 | Quadratic forms over global rings and fields |

11E25 | Sums of squares and representations by other particular quadratic forms |