FHBFACB. IX
whose complementary divisors are even or odd or are exact sth powers, and the excess of the sum of the kth powers of the divisors of the form 4m+1 of a number over the sum of the fcth powers of the divisors of the form 4m+3, as well as more technical sums of divisors defined on pages 297, 301-2, 305, 307-8, 314r-5 and 318. For the important case fc=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 u^.. .W&=TI 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 common divisor and least 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 functions. 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 equations, various small moduli, or miscellaneous methods.
Fermat expressed his belief that Fn = 22n-{-1 is a prime for every value of n. While this is true if n = 1, 2, 3, 4, it fails for n = 5 as noted by Euler. Later,