Elementary number theory.

*(English)*Zbl 0891.11001
Springer Undergraduate Mathematics Series. Berlin: Springer. xiv, 301 p. (1998).

Number theory is the story of mathematics. Its problems and its discoveries mark the stages of the history. It offers a concrete setting for explaining how mathematicians work: from experimental evidence through conjectures and counterexamples to proofs and applications. The flavour of even the deepest contemporary areas can often be presented without requiring too much mathematical background or maturity. So number theory is an appealing part of any mathematics course, and the authors have attempted to write a reader which can serve these ends at a number of levels.

The first part is pitched at first year level. It deals with the traditional topics: divisibility, primes, congruences and Euler’s function. The style is lucid, clearly organised with examples and graded exercises and surprisingly formal. Indeed, the proofs are intended to be read, and the layout is mercifully free of boxes and small type. The formal development dominates and there are only passing references to some of the historical figures and applications such as primality testing and factorisations and cryptography. The route that is chosen is the one which extends naturally to a treatment of Euclidean rings and unit groups, although the abstractions are not present at this stage.

The second part links further topics in number theory to second year algebra. Unit groups are examined, leading to primitive roots which are used in the work on quadratic residues. Quadratic reciprocity is proved by counting lattice points. The algebra of arithmetical functions is used to study the traditional examples.

The third part uses deeper mathematics. The chapter on Riemann’s \(\zeta (s)\) explains the connection with the primes, the analytic properties of the function and the importance of the Riemann Hypothesis. The chapter on sums of squares contains proofs of the 2 and 4 square theorems using properties of Gaussian integers and quaternions and a little geometry of numbers. Finally, a chapter on Fermat’s last theorem contains some of the highlights of the elementary theory and references to Wiles’ final resolution.

In line with the aims of the series, the book provides a careful introduction to number theory, but it seems likely that the teacher would wish to supplement it with material which picks up enriched applications. The solutions to the exercises are a strong point and very thorough.

The first part is pitched at first year level. It deals with the traditional topics: divisibility, primes, congruences and Euler’s function. The style is lucid, clearly organised with examples and graded exercises and surprisingly formal. Indeed, the proofs are intended to be read, and the layout is mercifully free of boxes and small type. The formal development dominates and there are only passing references to some of the historical figures and applications such as primality testing and factorisations and cryptography. The route that is chosen is the one which extends naturally to a treatment of Euclidean rings and unit groups, although the abstractions are not present at this stage.

The second part links further topics in number theory to second year algebra. Unit groups are examined, leading to primitive roots which are used in the work on quadratic residues. Quadratic reciprocity is proved by counting lattice points. The algebra of arithmetical functions is used to study the traditional examples.

The third part uses deeper mathematics. The chapter on Riemann’s \(\zeta (s)\) explains the connection with the primes, the analytic properties of the function and the importance of the Riemann Hypothesis. The chapter on sums of squares contains proofs of the 2 and 4 square theorems using properties of Gaussian integers and quaternions and a little geometry of numbers. Finally, a chapter on Fermat’s last theorem contains some of the highlights of the elementary theory and references to Wiles’ final resolution.

In line with the aims of the series, the book provides a careful introduction to number theory, but it seems likely that the teacher would wish to supplement it with material which picks up enriched applications. The solutions to the exercises are a strong point and very thorough.

Reviewer: J.H.Loxton (North Ryde)

##### MSC:

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

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

11Axx | Elementary number theory |

11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |

11D41 | Higher degree equations; Fermat’s equation |