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 <r(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 <r(x?) =y2 and <r(#2) =^ and John Wallis's problem to find solutions other than x =4 and y~5 of <r(xz)-<r(y2).
Fermat stated in 1640 that he had a proof of the fact, now known as Format's theorem, that, if p is any prime and x is any integer not divisible by p, then xp~l — 1 is divisible by p. This is one of the fundamental theorems 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 0(n) denotes the number of positive integers not exceeding n and relatively prime to n, then #*(ft)— 1 is divisible by n for every integer x relatively prime to w. Another elegant theorem states that, if p is a prime, 1+ \ 1-2-3 ---- (p— 1) I is divisible by p] it was first published 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 generalization of Wilson's theorem: if P denotes the product of the positive integers less than A and prime to A, then P+l is divisible by A if A =4, pm or 2pm, 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 N=*pii. . .p/«, where p1; ..., p, are distinct primes,
is divisible by JV, a fact due to Gauss for the case in which o is a prime.
Many cases have been found in which a71""1 — ! is divisible by n for a composite number n. But Lucas proved tlie following converse of Fermat's theorem: if a*— 1 is divisible by n when x =n — 1, but not when a; is a divisor < n — 1 of n— 1, then n is a prime.
Any integral symmetric function of degree d of 1, 2,..., p — 1 with integral coefficients is divisible by the prime p if d is not a multiple of p — 1. A generalization to the case of a divisor pn is due to Meyer (p. 101). Nielsen proved in 1893 that, if p is an odd prime and if k is odd and K k<p — 1, the sum of the products of 1, 2, . . ., p — 1 taken k at a time is divisible by p2. Taking k = p —2, we see that if p is a prime > 3 the numerator of the fraction