Reading Dickson

The efforts of Cantor and his collaborators show that a chronological history of mathematics down to the nineteenth century can be written in four large volumes. To cover the last century with the same elaborateness, it has been estimated that about fifteen volumes would be required, so extensive is the mathematical literature of that period. But to retain the chronological order and hence devote a large volume to a period of at most seven years would defeat some of the chief purposes of a history, besides making it very inconvenient to find all of the material on a particular topic. In any event there is certainly need of histories which treat of particular branches of mathematics up to the present time.

The theory of numbers is especially entitled to a separate history on account of the great interest which has been taken in it continuously through the centuries from the time of Pythagoras, an interest shared on the one extreme by nearly every noted mathematician, and on the other extreme by numerous amateurs attracted by no other part of mathematics. This history aims to give an adequate account of the entire literature of the theory of numbers. The first volume presents in twenty chapters the material relating to divisibility and primality. The concepts, results, and authors cited are so numerous that it seems appropriate to present here an introduction which gives for certain chapters an account in untechnical language of the main results in their historical setting, and for the remaining chapters the few remarks sufficient to clearly characterize the nature of their contents.

Since the history of each main topic is given separately, it has been possible without causing confusion to include reports on minor papers and isolated problems for the sake of completeness. In the cases of books and journals not usually accessible, the reports are quite full with some indication of the proofs. In other cases, proofs are given only when necessary to differentiate the paper from others deriving the same result.

The subject index contains a list of symbols; while $[x]$ usually denotes the greatest integer $\leqq x$, occasionally such square brackets are used to inclose an addition to a quotation. The symbol * before an author's name signifies that his paper was not available for report. The symbol † before a date signifies date of death. Initials are given only in the first of several immediately successive citations of an author.

The efforts of Cantor and his collaborators show that a chronological history of mathematics down to the nineteenth century can be written in four large volumes. To cover the last century with the same elaborateness, it has been estimated that about fifteen volumes would be required, so extensive is the mathematical literature of that period. But to retain the chronological order and hence devote a large volume to a period of at most seven years would defeat some of the chief purposes of a history, besides making it very inconvenient to find all of the material on a particular topic. In any event there is certainly need of histories which treat of particular branches of mathematics up to the present time.

The theory of numbers is especially entitled to a separate history on account of the great interest which has been taken in it continuously through the centuries from the time of Pythagoras, an interest shared on the one extreme by nearly every noted mathematician, and on the other extreme by numerous amateurs attracted by no other part of mathematics. This history aims to give an adequate account of the entire literature of the theory of numbers. The first volume presents in twenty chapters the material relating to divisibility and primality. The concepts, results, and authors cited are so numerous that it seems appropriate to present here an introduction which gives for certain chapters an account in untechnical language of the main results in their historical setting, and for the remaining chapters the few remarks sufficient to clearly characterize the nature of their contents.

Since the history of each main topic is given separately, it has been possible without causing confusion to include reports on minor papers and isolated problems for the sake of completeness. In the cases of books and journals not usually accessible, the reports are quite full with some indication of the proofs. In other cases, proofs are given only when necessary to differentiate the paper from others deriving the same result.

The subject index contains a list of symbols; while $\lfloor x\rfloor$ denotes the greatest integer $\le x$; Dickson writes $[x]$ for this, occasionally such square brackets are used to inclose an addition to a quotation. The symbol * before an author's name signifies that his paper was not available for report. The symbol † before a date signifies date of death. Initials are given only in the first of several immediately successive citations of an author.

A history organized by mathematical problems

Dickson's first volume, Divisibility and Primality (1919), contains twenty chapters. Its Preface, pp. iii–xii, also serves as its introduction. Chapter I occupies pp. 3–50 and treats perfect numbers, multiply perfect numbers, amicable numbers, and related divisor-sum questions.

A chronological history separates contributions to the same problem. Dickson instead keeps the topic together and follows successive contributions within it. Read his attribution as part of a mathematical statement: “Fermat stated” and “Euler proved” are different evidence, even when the displayed formula is identical.

For Chapter I, distinguish three questions. Does a formula produce a number with the desired property when its conditions hold? Do its conditions ever hold, or hold infinitely often? Does it exhaust all numbers with that property? Euclid's construction settles the first question. The converse for even perfect numbers settles a restricted version of the third. Neither statement answers the second.

What a reported theorem supplies

Dickson explains that he often omits proofs, especially when several authors reach the same result. An entry may therefore provide a statement, a reference, a calculation, or a proof sketch. Its brevity does not mean that a short proof has been given.

An elementary example separates these roles. If $n=ab$ with $a,b>1$, then

$$2^n-1=(2^a-1)(1+2^a+\cdots+2^{a(b-1)}),$$

so $2^n-1$ is composite. Thus primality of $n$ is necessary for $2^n-1$ to be prime. It is not sufficient: $2^{11}-1=23\cdot89$. Both points occur in Dickson's account, but the displayed factorization here supplies an argument beyond the reported statement.

In the following units, Modernized preserves the selected source's proof route and the status of its claims. Added proofs, missing-case analysis and regrouping of methods appear in Reinterpreted. Historical numerical lists retain their historical date; they are not lists of current records.

Conventions for reading the source

Dickson's asterisk before an author's name means that the paper was unavailable to him. It does not mean a result is false. A dagger before a date indicates death. Square brackets can mean either an integer part or an editorial addition to a quotation. A statement within a quotation is not automatically Dickson's endorsement.

All arithmetic below concerns positive integers unless a termination value of zero is explicitly introduced. A prime is an integer greater than 1 with exactly two positive divisors. “Coprime” means having greatest common divisor 1. Unique prime factorization is assumed. Geometric sums, divisor-sum multiplicativity, and the necessary congruence arguments are explained when used.

The readings follow the chapter's topics in order. Within Reinterpreted, an earlier calculation may be combined with a later explanation. For example, the even-perfect classification uses Dickson's short proof on p. 30 alongside Euclid and Euler on pp. 3 and 19.