414 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvin J. Perott9 applied the theory of commutative groups to show that, if <?!>> <?n are primes, there exist at least n 1 primes between qn and M*5i -«». T. J. Stieltjes10 expressed the product P of the primes 2, 3, . . ., p as a product AB of two factors in any way. Since A +B is not divisible by 2, . . . , p, there exists a prime >p. J. Hacks11 proved the existence of an infinitude f primes by use of his formula (Ch. XI, Hacks14) for the number of integers ^m not divisible by a square. C. 0. Boije af Gennas12 showed how to find a prime exceeding the nth prime pn>2. Take P = 2"13"> , . ,pn*n, each v^-l. Express P as a product of relatively prune factors 5, P/5, where Q = P/5 d > 1 . Since Q is divisible by no prime ^pn, it is a product of powers of primes &^pn+2. Take 5 so that Q< (p»-h2)2. Then Q is a prime. Axel Thue13 proved that, if (l+w)*<2n, there exist at least k+1 primes <2n. J. Braun13a noted that the sum of the inverses of the prunes ^p is, for p^. 5, an irreducible fraction > 1 ; hence the numerator contains at least one prime >p. He attributed to Hacks a proof by means of 11(1 l/p2)"^ 2s""2 = 7r2/6 ; the product would be rational if there were only a finite number of primes, whereas TT is irrational. E. Cahen14 proved the "identity of Euler" used by Kronecker.5 Stormer288 gave a proof. A. Le*vy15 took a product P of k of the first n primes pl9. . ., pn and the product Q of the remaining n k. Then P+Q is either prune or has a prime factor >pn; likewise for P Q. If pn is a prune such that pn+2 -is composite, there exist at least n prunes >pn, but ^l-f-pi£>2- -Pn- When Pl '" Pn is reduced to a simple fraction, the numerator has no factor in common with Pi . . .pn] hence there is a prime >pn. He considered (pp. 242-4) the primes defined by x(x 1) 1 for consecutive integers x. A. Auric16 assumed that p1}. . . , pk give all the primes. Then the number of integers < ?t =npia* is which is small in comparison with n, whence k increases indefinitely with n. 9Amer. Jour. Math., 11, 1888, 99-138; 13, 1891, 235-308, especially 303-5. "Annales fac. sc. de Toulouse, 4, 1890, 14, final paper. "Acta Math., 14, 1890-1, 335. "Ofversigt K. Sv. Vetenskaps-Akad. Forhand., Stockholm, 50, 1893, 469-471. "Archiv for Math, og Natur., Kristiania, 19, 1897, No. 4, 1-5. 13aDas Fortschreitungsgesetz der Primzahlen durch eine transcendente Gleichung exakt dargestellt, Wias. Beilage Jahresbericht, Gymn., Trier, 1899, 96 pp. "filaments de la the*orie des nombres, 1900, 319-322. "Bull, de Math, fil&nentaires, 15, 1909-10, 33-34, 80-82. 1&L'intenn6diaire des math., 22, 1915, 252.