It may seem to be a little late to be getting around to reviewing a book that was first published by McGraw-Hill in 1948 but it is still going more or less strong in Dover Publications’ series of classics in mathematics and science, so it is not out of place to give it a 21st century look.
Its author, whose first name is Øystein in his native Norwegian, lived from 1899 to 1968. His mathematical talent was such that he was lured to Yale in 1927 where he rose very rapidly through the ranks, becoming an associate professor in 1928, a full professor in 1929, and Sterling Professor in 1931. He made many contributions to algebra and has a theorem of graph theory named after him.
Besides his mathematical work Ore wrote good biographies of Abel and Cardano. His books display no traces of humor and I know of not a single anecdote about him. He was evidently one of those serious Scandinavian types, but I find him wholly admirable.
His book grew out of a course he offered at Yale that included more than a little bit of history. It goes slowly (for example, congruences do not appear until page 209) but it includes most of the topics that appear in current first courses in number theory. Some things are missing: Pell’s equation does not make it, nor do continued fractions, sums of four squares are not gone into, and there is no mention of the quadratic reciprocity theorem. On the other hand, it includes some things that are no longer given prominence, as Fermat’s and Euler’s methods of factorization, then more important then than they are now when integers can be factored with the touch of a button. There is a proof that a regular 17-gon can be constructed with straightedge and compass. I never knew, before reading it on page 107, that
( [a, b], [a, c], [b, c] ) = [ (a, b), (a, c), (b, c) ]
(the round brackets denote greatest common divisor and the square brackets, least common multiple). Being an algebraist, Ore cannot resist introducing groups, rings, fields, lattices, and modules (which he spells “moduls”) but he does not belabor them.
History appears throughout, at greater length than the historical notes sometimes found in other texts. It is familiar to those to whom it is familiar, but even so there are surprises: Mersenne was an “aggressive theologian”. To those encountering the history for the first time, it should be a rewarding feature of the book.
It is to be expected that a book in its seventh decade of life should be showing some signs of age. We no longer write “algorism” or “decadic” and “aliquot parts” is not in our active vocabularies but there is no difficulty in knowing what the author means. The numerical results now sound quaint: the Goldbach conjecture is true up to 100,000, there is no odd perfect number less than 2,000,000, the largest known Mersenne prime is “this huge number of 39 digits”. The references are no longer as helpful as they were in 1948.
Ore includes problems but many of them are routine numerical exercises and there are no answers or solutions. For that reason, and because of the book’s age, it is probably not suitable for people trying to learn number theory by themselves.
So, what should we do with it? We should give it as a present, wrapped with another more modern introduction to number theory (Dover publishes quite a few of those, at Dover’s always low prices). Number theorists should own a copy, as homage to a master if nothing else. Non-number theorists should take a look at it: it is good and easy reading and they might learn something interesting. I hope it survives to 2048, or longer.
Woody Dudley immodestly suggests his Elementary Number Theory (Dover edition 2008) as a supplement.