418 HlSTOKY OP THE THEORY OF NUMBEKS. [CHAP. XVIII H. Weber87 and E. Sobering68 completed Dirichlet's85 proof of his first theorem. A. Meyer59 completed Dirichlet's56 proof of his extended theorem. F. Mertens60 gave an elementary proof of Dirichlet's66 extended theorem. Ch. de la Valtee-Poussin86 proved that the number of primes £x repre-sentable by a properly primitive definite positive or indefinite61 irreducible binary quadratic form is asymptotic to gx/logx, where g is a constant; and the same for primes belonging also to a linear form compatible with the character of the quadratic form. L. Kronecker5 (pp. 494-5) stated a theorem on factorable forms in several variables which represent an infinitude of primes. ELEMENTARY PROOFS OF THE EXISTENCE OF AN INFINITUDE OF PRIMES mz-\~l, FOR ANY GIVEN m. V. A. Lebesgue66 gave a proof for the case m a prime, using the fact that xm~1—xm~2y+. . .+ym~l has besides the possible factor m only prime factors 2km+l. A like method applies65a to 2mz— 1. J. A. Serret66 gave an incomplete proof for any m. F. Landry67 gave- a proof like Lebesgue's.65 If 0 is the largest prime 2km +1 and if x is the product of all of them, xm+l is divisible by no one of them. Since (a;m+l)/(ic+l) has no prime divisor not of the form 2km+l, there exists at least one >0. A. (ienocchi68 proved the existence of an infinitude of primes mz=± 1 and n*z=fcl for n a prime by use of the rational and irrational parts of L. Kronecker5 (pp. 440-2) gave in lectures, 1875-6, a proof for the case m a prime; the simple extension in the text to any m was added by Hensel. E. Lucas gave a proof by use of his un (Lucas,39 p. 291, of Ch. XVII). A. Lefe*bure30 of Ch. XVI stated that the theorem follows from his results. L. Kraus69 gave a proof. A. S. Bang70 and Sylvester83 proved it by use of cyclotomic functions. K. Zsigmondy78 of Ch. VII gave a proof. Also, E. Wendt,71 and Birkhoff and Vandiver62 of Ch. XVI. l7Math. Annalen, 20, 1882, 301-329. Elliptische Functionen (= Algebra, III), ed. 2, 1908, 613-6. "Werke, 2, 1909, 357-365, 431-2. "Jour, fiir Math., 103, 1888, 98-117. Exposition by Bachmann,84 pp. 272-307. "Sitzungsber. Ak. Wiss. Wien (Math.), 104, 1895, Ha, 1093-1153, 1158. Simplification, ibid., 109, 1900, Ha, 415-480. 61Cf. E. Landau, Jahresber. D. Math. Verein., 24, 1915, 250-278. MJour. de Math., 8, 1843, 51, note. Exercices d'analyse nume'rique, 1859, 91. M°Jour. de Math., (2), 7, 1862, 417. MJour. de Math., 17, 1852, 186-9. 67Deuxieme me'moire sur la th6orie des nombres, Paris, 1853, 3. «8Annali di mat., (2), 2, 1868-9, 256-7. Cf. .Genocchi22- 61 of Ch. XVII. 69Casopis Math, a Fys., 15, 1886, 61-2. Cf. Fortschritte, 1886, 134-5. 70Tidflskrift for Math., (5), 4, 1886, 70-80, 130-7. See Bang". « Ch. XVI. 71 Jour, fiir Math., 115, 1895, 85.