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Introduction to Number Theory

Trygve Nagell
AMS Chelsea
Publication Date: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a straightforward introduction to elementary number theory, that doesn’t require anything beyond high school math and that covers the classical results. The present volume was originally published in 1951 by Wiley and then reprinted by Chelsea in 1964. It is especially strong on Diophantine equations (the author’s research specialty). Another unusual feature is elementary proofs of Dirichlet’s theorem on primes in arithmetic progressions for several special cases. For example, there’s a well-known elementary proof that there are infinitely many primes of the form \(4n - 1\), but this book proves a number of other special cases. It was probably the first elementary book to include Selberg’s elementary proof of the Prime Number Theorem, although it follows Selberg’s original (and difficult) proof very closely. The book also has an enormous number of exercises, many of them difficult (there are no solutions, though).

There were a number of similar textbooks published in the first half of the twentieth century, such as Vinogradov’s Elements of Number Theory. Each of them has its specialties and strengths. Since 1950 not much has come out along these lines, although a good modern version is Rassias’s 2011 Problem-Solving and Selected Topics in Number Theory. It covers roughly the same ground but has some new proofs; it also has a large collection of new worked problems, many taken from Math Olympiads, journal problem columns, and the Putnam exams.

These books are not as comprehensive as the “big gun” introductions, such as Hardy & Wright’s An Introduction to the Theory of Numbers, or Niven & Zuckerman & Montgomery’s An Introduction to the Theory of Numbers. They also do not use algebraic language or approaches, as do the more advanced introductions such as Rose’s A Course in Number Theory or Ireland & Rosen’s A Classical Introduction to Modern Number Theory.


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Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

  • Divisibility: 1.1 Divisors; 1.2 Remainders; 1.3 Primes; 1.4 The fundamental theorem; 1.5 Least common multiple and greatest common divisor; 1.6 Moduls, rings and fields; 1.7 Euclid’s algorithm; 1.8 Relatively prime numbers. Euler’s \(\varphi\)-function; 1.9 Arithmetical functions; 1.10 Diophantine equations of the first degree; 1.11 Lattice points and point lattices; 1.12 Irrational numbers; 1.13 Irrationality of the numbers \(e\) and \(\pi\); Exercises (1--40)
  • On the Distribution of Primes: 2.14 Some lemmata; 2.15 General remarks. The sieve of Eratosthenes; 2.16 The function \(\pi(x)\); 2.17 Some elementary results on the distribution of primes; 2.18 Other problems and results concerning primes
  • Theory of Congruences: 3.19 Definitions and fundamental properties; 3.20 Residue classes and residue systems; 3.21 Fermat’s theorem and its generalization by Euler; 3.22 Algebraic congruences and functional congruences; 3.23 Linear congruences; 3.24 Algebraic congruences to a prime modulus; 3.25 Prime divisors of integral polynomials; 3.26 Algebraic congruences to a composite modulus; 3.27 Algebraic congruences to a prime- power modulus; 3.28 Numerical examples of solution of algebraic congruences; 3.29 Divisibility of integral polynomials with regard to a prime modulus; 3.30 Wilson’s theorem and its generalization; 3.31 Exponent of an integer modulo \(n\); 3.32 Moduli having primitive roots; 3.33 The index calculus; 3.34 Power residues. Binomial congruences; 3.35 Polynomials representing integers; 3.36 Thue’s remainder theorem and its generalization by Scholz; Exercises (41-89)
  • Theory of Quadratic Residues: 4.37 The general quadratic congruence; 4.38 Euler’s criterion and Legendre’s symbol; 4.39 On the solvability of the congruences \(x^2\equiv\pm 2\, (\text{mod }p)\); 4.40 Gauss’s lemma; 4.41 The quadratic reciprocity law; 4.42 Jacobi’s symbol and the generalization of the reciprocity law; 4.43 The prime divisors of quadratic polynomials; 4.44 Primes in special arithmetical progressions
  • Arithmetical Properties of the Roots of Unity: 5.45 The roots of unity; 5.46 The cyclotomic polynomial; 5.47 Irreducibility of the cyclotomic polynomial; 5.48 The prime divisors of the cyclotomic polynomial; 5.49 A theorem of Bauer on the prime divisors of certain polynomials; 5.50 On the primes of the form \(n y-1\); 5.51 Some trigonometrical products; 5.52 A polynomial identity of Gauss; 5.53 The Gaussian sums; Exercises (90-122)
  • Diophantine Equations of the Second Degree: 6.54 The representation of integers as sums of integral squares; 6.55 Bachet’s theorem; 6.56 The Diophantine equation \(x^2-Dy^2=1\); 6.57 The Diophantine Equation \(x^2-Dy^2=-1\); 6.58 The Diophantine equation \(u^2-Dv^2=C\); 6.59 Lattice points on conics; 6.60 Rational points in the plane and on conics; 6.61 The Diophantine equation \(ax^2+by^2+cz^2=0\)
  • Diophantine Equations of Higher Degree: 7.62 Some Diophantine equations of the fourth degree with three unknowns; 7.63 The Diophantine equation \(2x^4-y^4=z^2\); 7.64 The quadratic fields \(K(\sqrt{-1}), K(\sqrt{-2})\) and \(K(\sqrt{-3})\); 7.65 The Diophantine equation \(\xi^3+\eta^3+\zeta^3=0\) and analogous equations; 7.66 Diophantine equations of the third degree with an infinity of solutions; 7.67 The Diophantine equation \(x^7+y^7+z^7=0\); 7.68 Fermat’s last theorem; 7.69 Rational points on plane algebraic curves. Mordell’s theorem; 7.70 Lattice points on plane algebraic curves. Theorems of Thue and Siegel; Exercises (123-171)
  • The Prime Number Theorem: 8.71 Lemmata on the order of magnitude of some finite sums; 8.72 Lemmata on the Möbius function and some related functions; 8.73 Further lemmata. Proof of Selberg’s formula; 8.74 An elementary proof of the prime number theorem; Exercises (172-180)
  • Table of primitive roots
  • Fundamental solutions of equations \(x^2-Dy^2=\pm 1\)
  • Name index
  • Subject index