PREFACE. IX 
whose complementary divisors are even or odd or are exact sth powers, and 
the excess of the sum of the A;th powers of the divisors of the form 4m +1 of 
a number over the sum of the A;th powers of the divisors of the form 4m -f 3, 
as well as more technical sums of divisors defined on pages 297, 301-2, 305, 
307-8, 314-5 and 318. For the important case k = 0, such a sum becomes 
the number of the divisors in question. There are theorems on the number 
of sets of positive integral solutions of UiU^. . .Uk = n or of x''y^ = n. Also 
Glaisher's cancellation theorems on the actual divisors of numbers (pp. 
^' 310-11, 320-21). Scattered through the chapter are approximation and 
asymptotic formulas involving some of the above functions. 
In Chapter XI occur Dirichlet's theorem on the number of cases in the 
division of n by 1, 2, . . . , p in turn in which the ratio of the remainder to the 
divisor is less than a given proper fraction, and the generalizations on pp. 
330-1; theorems on the number of integers ^n which are divisible by no 
,( exact sth power > 1 ; theorems on the greatest divisor which is odd or has 
/ specified properties; many theorems on greatest coromon divisor and least 
I common multiple ; and various theorems on mean values and probability. 
' The casting out of nines or of multiples of 11 or 7 to check arithmetical 
computations is of early origin. This topic and the related one of testing 
the divisibility of one number by another have given rise to the numerous 
elementary papers cited in Chapter XII. 
The frequent need of the factors of numbers and the excessive labor 
required for their direct determination have combined to inspire the 
construction of factor tables of continually increasing limit. The usual 
method is essentially that given by Eratosthenes in the third century B. C. 
A special method is used by Lebon (pp. 355-6). Attention is called to 
Lehmer's Factor Table for the First Ten Millions and his List of Prime 
Numbers from 1 to 10,006,721, published in 1909 and 1914 by the Carnegie 
Institution of Washington. Since these tables were constructed anew with 
the greatest care and all variations from the chief former tables were taken 
account of, they are certainly the most accurate tables extant. Absolute 
accuracy is here more essential than in ordinary tables of continuous func- 
tions. Besides giving the history of factor tables and lists of primes, this 
Chapter XIII cites papers which enumerate the primes in various intervals, 
prime pairs (as 11, 13), primes of the form 4n+l, and papers listing primes 
written to be base 2 or large primes. 
Chapter XIV cites the papers on factoring a number by expressing it as 
a difference of two squares, or as a sum of two squares in two ways, or by use 
of binary quadratic forms, the final digits, continued fractions. Pell equa- 
tions, various small moduli, or miscellaneous methods. 
Fermat expressed his belief that Fn = 2^"+l is a prime for every value of n. 
While this is true if n = 1, 2, 3, 4, it fails forn = 5 as noted by Euler. Later, 
