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Publisher:

Dover Publications

Publication Date:

2005

Number of Pages:

486

Format:

Paperback

Price:

25.95

ISBN:

0486442322

Category:

Monograph

The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by , on ]

Fernando Q. Gouvêa

03/22/2007

Leonard Eugene Dickson's three volume *History of the Theory of Numbers* is an indispensable reference. It is old enough to be in the public domain, and the result is that it is now available in two different editions: three hardcover volumes from AMS/Chelsea and three softcover volumes from Dover. The first volume deals with *Divisibility and Primality* (this is the page for the Dover edition; go here for the AMS/Chelsea edition). The second (and by far the largest) volume deals with *Diophantine Analysis* (Dover edition, Chelsea edition). The third covers *Quadratic and Higher Forms* (Dover edition, Chelsea edition). This review covers all three volumes.

The first thing to say is that Dickson's *History* is not really a history at all. It is more like a huge annotated bibliography. In each area, Dickson lists every paper he could find on the subject and describes what each achieved. In some cases he gives details, but in many cases only the bare result is given. No serious attempt is made to trace influences, historical connections or context, or even to analyze the conceptual development of the field. Thus, this volume is a sort of predecessor of the massive compilations of *Reviews in Number Theory* selected by the AMS from the *Mathematical Reviews* database.

The three volumes were originally published in 1919, 1920, and 1923. Some of the story behind them is told in a valuable article by Della Dumbaugh Fenster, "Why Dickson Left Quadratic Reciprocity Out of His *History of the Theory of Numbers*" (*American Mathematical Monthly*, **106** (1999), pp. 618–627).

As to Dickson's motivation and approach, Fenster points to a passage from the introduction to volume 2 that is worth quoting at length:

…What is generally wanted [in a history that may be useful to a researcher in the field] is a full and correct statement of the facts, not an historian's personal explanation of those facts. The more completely the historian remains in the background or the less conscious the reader is of the historian's personality, the better the history. Before writing such a history, he must have made a more thorough search for all the facts than is necessary for the conventional history. With such a view of the ideal self-effacement of the historian, what induced the author to interrupt his own investigations for the greater part of the past nine years to write this history? Because it fitted in with his conviction that every person should aim to perform at some time in his life some serious, useful work for which it is highly improbable that there will be any reward whatever other than his satisfaction therefrom. (Volume 2, pages xx-xxi.)

Thus, Dickson was really trying to produce a catalog of "facts," and that is what he achieved. He viewed this as a service to those doing research in the field, and it is clear that the book was indeed useful in this way. Nowadays, it is probably most useful to historians. In fact, as D. N. Lehmer said in a review of the first edition, it is "a list of references from which a history of the theory of numbers might be written." Such a history of the whole of number theory is probably inconceivable. (How many volumes would it have?) Nevertheless, these volumes are an indispensable starting point for anyone who wants to find out more about the history of number theory up to about 1920.

As the title of Fenster's article highlights, there is one huge gap: quadratic reciprocity and more general reciprocity laws are not mentioned at all. This seems to have been at least partly due to Dickson's running out of steam: in the third volume, he announces that there will be a fourth volume dealing with those topics, but that volume never appeared. A Ph.D. thesis by one of Dickson's students seems to have been intended as a chapter in that fourth volume, and the student, Albert Everett Cooper, seems to have written more than that one chapter. None of this work ever appeared in print, perhaps because no publisher could be found. Fenster finds little evidence that Dickson pressed very hard for publication, however.

It may well be that there was a further reason. In the 1920s, the search for "the most general reciprocity law," now embedded in what became known as Class Field Theory, was nearing its climax. Class Field Theory, however, was a huge undertaking. Dickson may well have felt that a volume on reciprocity that did not include the work of Artin, Takagi, Furtwängler, and others would be inadequate, but that including that work would so enlarge the scope of the work as to make it impossible to complete.

It would be inconceivable, today, for a single scholar to attempt to survey the whole of number theory. Even in Dickson's day, it was a very ambitious thing to attempt. That Dickson did as well as he did is quite impressive, and the volumes he produced continue to be essential to anyone interested in number theory and its history.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He believes, *contra* Dickson, that good history *is* "the historian's personal explanation" of the facts.

I. Perfect, multiply perfect, and amicable numbers | |||||||

II. Formulas for the number and sum of divisors, problems of Fermat and Wallis | |||||||

III. Fermat’s and Wilson’s theorems, generalizations and converses; symmetric functions of 1, 2, ..., p-1, modulo p | |||||||

IV Residue of (up-1-1)/p modulo p | |||||||

V. Euler’s function, generalizations; Farey series | |||||||

VI. Periodic decimal fractions; periodic fractions; factors of 10n | |||||||

VII. Primitive roots, exponents, indices, binomial congruences | |||||||

VIII. Higher congruences | |||||||

IX. Divisibility of factorials and multinomial coefficients | |||||||

X. Sum and number of divisors | |||||||

XI. Miscellaneous theorems on divisibility, greatest common divisor, least common multiple | |||||||

XII. Criteria for divisibility by a given number | |||||||

XIII. Factor tables, lists of primes | |||||||

XIV. Methods of factoring | |||||||

XV. Fermat numbers | |||||||

XVI. Factors of an+bn | |||||||

XVII. Recurring series; Lucas’ un, vn | |||||||

XVIII. Theory of prime numbers | |||||||

XIX. Inversion of functions; Möbius’ function; numerical integrals and derivatives | |||||||

XX. Properties of the digits of numbers | |||||||

Indexes |

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