PREFACE. V 
Interesting amicable triples and amicable numbers of higher order have 
been recently found by Dickson and Poulet (p. 50). 
Although it had been employed in the study of perfect and amicable 
numbers, the explicit expression for the sum a{n) of all the divisors of n is 
reserved for Chapter II, in which is presented the history of Fermat's two 
problems to solve (T(x^)=y^ and <t{x^) =y^ and John Wallis's problem to find 
solutions other than a; = 4 and y = 5 oi (T{x^)=o-{y^). 
Fermat stated in 1640 that he had a proof of the fact, now known as 
Fermat's theorem, that, if p is any prime and x is any integer not divisible 
by p, then x^~^ — l is divisible by p. This is one of the fundamental theo- 
rems of the theory of numbers. The case x = 2 was known to the Chinese as 
early as 500 B. C. The first published proof was given by Euler in 1736. 
Of first importance is the generalization from the case of a prime p to any 
integer n, published by Euler in 1760: if (/)(n) denotes the munber of positive 
integers not exceeding n and relatively prime to n, then x*^"^ — 1 is divisible 
. by n for every integer x relatively prime to n. Another elegant theorem 
states that, if p is a prime, l+jl-2-3. . . .{p — l)\ is divisible by p; it was 
first pubUshed by Waring in 1770, who ascribed it to Sir John Wilson. This 
theorem was stated at an earlier date in a manuscript by Leibniz, who with 
Newton discovered the calculus. But Lagrange was the first one to publish 
(in 1773) a proof of Wilson's theorem and to observe that its converse is 
true. In 1801 Gauss stated and suggested methods to prove the generali- 
zation of Wilson's theorem: if P denotes the product of the positive integers 
less than A and prime to A, then P+1 is divisible by A if A =4, p"" or 2p"*, 
where p is an odd prime, while P — 1 is divisible by A if A is not of one of 
these three forms. A very large number of proofs of the preceding theorems 
are given in the first part of Chapter III. Various generalizations are then 
presented (pp. 84-91). For instance, if iV = p/' . . . p/*, where Pi, ..., p« 
are distinct primes, 
a^-(a^/P'+ . . . +a^/PO + (a^/P'P'+ ...)-••• . +(-l)''a^/^---P'' 
is divisible by N, a fact due to Gauss for the case in which a is a prime. 
Many cases have been found in which o"~^ — 1 is divisible by n for a 
composite number n. But Lucas proved the following converse of Fermat's 
theorem : if a^ — 1 is divisible by n when x = n — l, but not when x is a divisor 
|<n — 1 of 71 — 1, then w is a prime. 
Any integral symmetric function of degree d of 1, 2, . . ., p — 1 with 
I integral coefficients is divisible by the prime p if c^ is not a multiple of p — 1. 
A generalization to the case of a divisor p" is due to Meyer (p. 101) . Nielsen 
proved in 1893 that, if p is an odd prime and if k is odd and l<fc<p — 1, the 
sum of the products of 1, 2, . . ., p — 1 taken A; at a time is divisible by p^. 
Taking fc = p — 2, we see that if p is a prime > 3 the numerator of the fraction 
A 
