X PREFACE. 
Gauss proved that a regular polygon of m sides can be constructed by ruler 
and compasses if m is a product of a power of 2 and distinct odd primes each 
of the form Fn, and stated correctly that the construction is impossible if m 
is not such a product. In view of the papers cited in Chapter XV, F„ is 
composite if n = 5, 6, 7, 8, 9, 11, 12, 18, 23, 36, 38 and 73, while nothing is 
known for other values >4 of n. No conoment will be made on the next 
chapter which treats of the factors of numbers of the form o"±6" and of 
certain trinomials. 
In Chapter XVII are treated questions on the divisors of terms of a 
recurring series and in particular of Lucas' functions 
a — o 
where a and h are roots oi x'^ — Px-\-Q = Q, P and Q being relatively prime 
integers. By use of these functions, Lucas obtained an extension of Euler's 
generaUzation of Fermat's theorem, which requires the correction noted by 
Carmichael (p. 406), as well as various tests for primality, some of which 
have been emploj^ed in investigations on perfect numbers. Many papers on 
the algebraic theory of recurring series are cited at the end of the chapter. 
Euchd gave a simple and elegant proof that the number of primes is infi- 
nite. For the generalization that every arithmetical progression n, n+m, 
n-\-2m,. . ., in which n and m are relatively prime, contains an infinitude 
of primes, Legendre offered an insufficient proof, while Dirichlet gave his 
classic proof by means of infinite series and the classes of binary quadratic 
forms, and extended the theorem to complex integers. Mertens and others 
obtained simpler proofs. For various special arithmetical progressions, the 
theorem has been proved in elementary ways by many writers. Dirichlet 
also obtained the theorems that, if a, 26, and c have no common factor, 
ax'^+2hxy-\-cy^ represents an infinitude of primes, while an infinitude of these 
primes are representable by any given linear form Mx+N with M and N 
relatively prime, pro\^ded a, h, c, M, N are such that the quadratic and linear 
forms can represent the same number. 
No complete proof has been found for Goldbach's conjecture in 1742 that 
every even integer is a sum of two primes. One of various analogous un- 
proved conjectures is that every even integer is the difference of two consec- 
utive primes in an infinitude of ways (in particular, there exists an infinitude 
of pairs of primes differing by 2). No comment will be made on the further 
topics of this Chapter XVIII: polynomials representing numerous primes, 
primes in arithmetical progression, tests for primality, number of primes 
between assigned limits, Bertrand's postulate of the existence of at least one 
prime between x and 2x — 2 for x>3, miscellaneous results on primes, 
diatomic series, and asymptotic distribution of primes. 
