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Elementary theory of numbers. Translated from the Polish by A. Hulanicki. (English) Zbl 0122.04402

Monografie Matematyczne. Tom 42. Warszawa: Państwowe Wydawnictwo Naukowe. 480 p. (1964).
This work is evidently intended for a class of readers whose acquaintance with mathematics is slight, so that the term “elementary” appearing in the title can, for once, be taken entirely at face value. No advanced analytic or algebraic techniques are used and there is no reason why a keen and gifted schoolboy should not follow every step of the very clear and careful exposition. Where proofs are supplied, they are presented in considerable detail; but in cases where the argument is judged to be too exacting, theorems are stated without proof. The discussion is enlivened by innumerable references to unsolved problems and conjectures and made more concrete by copious numerical illustrations. The amount of information contained in these pages is large, though it must be noted that the structure of the book is rather loose: the individual chapters are almost independent of each other and the sequence of topics does not seem to follow a recognizable pattern. Nor is the reader always able to gauge the relative importance of various results discussed by the author.
The contents of the book can be gathered from the following chapter headings supplemented by brief comments.
I. ‘Divisibility and indeterminate equations of first degree’.
II. ‘Diophantine analysis of second and higher degrees’. Here the author investigates a large number of special diophantine equations of degrees 2, 3 or 4, such as \(x^2 + y^2 = z^2\) and \(x^4 + y^4 = z^2\). He also treats Pell’s equation.
III. ‘Prime numbers’. Additive properties of prime numbers are discussed, as are also questions relating to primes in progressions and the representation of primes by polynomials. Full proofs are given of Bertrand’s postulate and of Chebyshev’s theorem on the order of magnitude of \(\pi(n)\).
IV. ‘Number of divisors and their sum’. Some elementary properties of the functions \(d(n)\) and \(\sigma(n)\) are discussed, their average asymptotic behaviour is investigated, and the Dirichlet series associated with them are expressed in terms of the zeta function.
V. ‘Congruences’. This chapter contains, for example, results on roots of polynomials, proofs of Wilson’s and Fermat’s theorems, and the study of the representation of primes of the form \(4k+1\) as sums of two squares.
VI. ‘Euler’s totient function and the theorem of Euler’. Simple arithmetic properties and the rate of growth of \(\varphi(n)\) are studied, Euler’s generalization of Fermat’s theorem is established, and it is shown that there exist infinitely many primes congruent to 1 modulo \(k\). This is followed by a discussion of primitive roots, indices, and \(n\)-th power residues.
VII. ‘Representation of numbers by decimals in a given ‘Continued fractions’. This includes, in particular, results on the approximation of real by rational numbers, and also applications to Pell’s equation.
IX. ‘Legendre’s symbol and Jacobi’s symbol’. The principal topic considered here is the law of quadratic reciprocity.
X. ‘Mersenne numbers and Fermat numbers’. This chapter contains much miscellaneous information and, in particular, the tests of primality of Mersenne numbers due to Lucas and D. H. Lehmer.
XI. ‘Representations of natural numbers as sums of non-negative \(k\)-th powers’. The author discusses the representation of integers as sums of 2, 3 or 4 squares or cubes. He shows that every natural number is the sum of 50 biquadrates, and considers briefly the general Waring Problem.
XII. ‘Some problems of the additive theory of numbers’. This is rather a mixed collection. Generating functions of partition theory are introduced, magic squares are discussed, results such as K. F. Roth’s theorem on progression-free sets are mentioned, and there is a proof of Schur’s theorem to the effect that if the natural numbers are divided in any manner into a finite number of classes, then the equation \(x + y = z\) is soluble in at least one class.
XIII. ‘Complex integers’. This chapter develops the elementary theory of Gaussian integers and their divisibility properties. These ideas are applied to the determination of the number of representations of n as the sum of 2 or 4 squares.
There is an excellent bibliography, an author index, and a subject index.
Reviewer: Leon Mirsky

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11Axx Elementary number theory
11D04 Linear Diophantine equations
11D09 Quadratic and bilinear Diophantine equations
11D25 Cubic and quartic Diophantine equations
11P81 Elementary theory of partitions
05B15 Orthogonal arrays, Latin squares, Room squares
Full Text: EuDML