Invitation to number theory Book

A charming and very readable introduction to number theory perfectly pitched for readers with intermediate high school knowledge of mathematics (a basic knowledge of divisibility and of Pythagoras´ theorem is probably sufficient to carry the reader through). The editors state that: "This book is one of a series written by professional mathematicians in order to make some important mathematical ideas interesting and readable to a large audience of high school students and laymen." Measured by this goal, I believe the book is wonderfully successful and hence the five stars in my rating.

The text niftily and succintly weaves some interesting history, beguiling mathematical problems and, in general, clear proofs of key theorems and assertions together with a smattering of problems to induce the reader to work out and understand the a well chosen set of related mathematical concepts. The books is an entertaining stroll which leads up to the solutions of the Pythagorean equation (for which integer values of the three sides are there right-angled triangles?) and to Fermat´s congruence (also known as the little Fermat´s theory) but pointing out important features on the way such as magic squares, Mersenne and Fermat primes, perfect and amicable numbers, numeration systems, the algebra of congruences and its application to checks on computations, finding out the day of the week any past or future date falls and how to build round-robin tournament schedules, amongst others. Written in 1967, it is a bit dated as regards some aspects of the field such as, understandably, the largest primes calculated using computers and I´m sure that, were Professor Ore (1899-1968) alive today and updating the book, he would carefully add a little something about the applications of number theory to cryptography. However, this does not detract from the merits of this delightful gem of a book.

It is worth recalling the editors´ key recommendation, the reader "should keep in mind that a book on mathematics cannot be read quickly. Nor must he expect to understand all parts of the book on first reading. He should feel free to skip complicated parts and return to them later; often an argument will be clarified by a subsequent remark". Finally they are perfectly right in stating that "The best way to learn mathematics is to do mathematics" and to that extent the problems set by the author are satisfactory.