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, Fn 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 comment will be made on the next chapter which treats of the factors of numbers of the form an=±=6n 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
where a and 5 are roots of z2-Pz-f-Q = 0, P and Q being relatively prime integers. By use of these functions, Lucas obtained an extension of Euler's generalization 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 employed in investigations on perfect numbers. Many papers on the algebraic theory of recurring series are cited at the end of the chapter.
Euclid gave a simple and elegant proof that the number of primes is infinite. For the generalization that every arithmetical progression n, n+m, n+2w,. . ., 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*+2bxy+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, provided a, fe, c, M, N are such that the quadratic and lineas 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 unproved conjectures is that every even integer is the difference of two consecutive 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.