VI PREFACE. 
equal to 1 + 1/2+1/3+ . . . +l/(p — 1) is divisible by p'^, a result first proved 
by Wolstenholme in 1862. Sylvester stated in 1866 that the sum of all 
products of n distinct numbers chosen from 1, 2, . . . , w is divisible by each 
prime > n + 1 which is contained in any term of the set m — n + 1 , . . . , w, m + 1 . 
There are various theorems analogous to these. 
In Chapter IV are given properties of the quotient {uP~^ — l)/p, which 
plays an important role in recent investigations on Fermat's last theorem 
(the impossibiUty of x'^-\-y^ = z^ if p>2), the history of which will be treated 
in the final chapter of Volume II. Some of the present papers relate to 
(w*^"^ — 1)/«, where n is not necessarily a prime. 
TMiile Euler's ^-function was defined above in order to state his general- 
ization of Fermat's theorem, its numerous properties and generalizations 
are reserved for the long Chapter V. In 1801 Gauss gave the result that 
4>{d^ + . . . -\-<i>{dk) = n, if di, . . . , d^ are the divisors of n; this was generalized 
by Laguerre in 1872, H. G. Cantor in 1880, Busche in 1888, Zsigmondy in 
1893, Vahlen in 1895, Elliott in 1901, and Hammond in 1916. In 1808 
Legendre proved a simple formula for the number of integers ^ n which are 
divisible by no one of any given set of primes. The asymptotic value of 
(f>{l)-\- . . . +0(G) for G large was discussed by Dirichlet in 1849, Mertens in 
1874, Perott in 1881, Sylvester in 1883 and 1897, Cesaro in 1883 and 1886-8, 
Berger in 1891, and Kronecker in 1901. The solution of 4>{x)=g was treated 
by Cayley in 1857, Mmin in 1897, Pichler in 1900, Carmichael in 1907-9, 
Ranum in 1908, and Cunningham in 1915. H. J. S. Smith proved in 1875 
that the m-rowed determinant, ha\ing as the element in the ith row and 
ji\i column any function fib) of the greatest common divisor 5 of i and j, 
equals the product of F{\), F{2),. . ., F(m), where 
F(m)=/(m)-2/g)+2/(^J-...., m = py 
In particular, F{m)=<t>{m) if f{8)=8. In several papers (pp. 128-130) 
Cesaro considered analogous determinants. The fact that 30 is the largest 
number such that all smaller numbers relatively prime to it are primes was 
first proved by Schatunowsky in 1893. 
A. Thacker in 1850 evaluated the sum 4>k{n) of the kth. powers of the 
integers ^n which are prime to n. His formula has been expressed m^ 
various symbolic forms by Ces^o and generalized by Glaisher and Nielsen./ 
Crelle had noted in 1845 that <piin) = |n0( n). In 1869 Schemmel considered 
the number of sets of n consecutive integers each < m and prime to m. In 
connection with linear congruence groups, Jordan evaluated the number of 
different sets of k positive integers ^?i whose greatest common divisor is 
prime to n. This generalization of Euler's (^-function has properties as 
simple as the latter function and occurs in many papers under a variety of 
notations. It in turn has been generalized (pp. 151-4). 
