134 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. V Rogel87 considered the number of integers v<n such that v and n are not both divisible by the rth power of a prime. Also the number when each prune factor common to v and n occurs in them exactly to the rth power. I. T. Kaplan published at Odessa in 1897 a pamphlet in Russian on the distribution of the numbers relatively prime to a given number. M. Bauer88 proved that, for x prime to m, kx+l represents iKdA) 4>(m) W integers relatively prime to m and incongruent modulo m, where di is the g. c. d. (k, m) of k, m, and d2 = (Z, m), (dlf d2) = 1, while is the number of incongruent integers prime to m = p^1 . . . pae* which are represented by kx-\~l when k, I, x are prime to m. Of those integers, \fr(m)/\l/(pi. . .pr) are divisible only by the special prime factors plt . . ., pr of m. J. de Vries88a proved the first formula of Dirichlet's.21 C. Moreau89 evaluated cj>(ri) by the method of Grunert.15 E. Landau90 proved that where e is of the order of magnitude of or1 log x, C is Euler's constant, and f is Riemann's f -function. P. Wolfskehl91 proved by Tcheby chef's theorem that the $(ri) integers <n and prime to n are all primes only when n = 1, 2, 3, 4, 6, 8, 12, 18, 24, 30. [Schatunowsky.75] E. Landau92 gave a proof, without the use of Tchebychef s theorem, by finding a lower limit to the number of integers k having no square factor >1, where t^k> 5i/8. E. Maillet,93 by use of Tchebychef 's theorem, proved the same result and the generalization: Given any integer r, there exist only a finite number of integers N such that the <t>(N) integers <N and relatively prime to N contain at most r equal or distinct prime factors. Alois Pichler94 noted that $(x) ~n has no solution if n is odd and >1; while^ (x) =2* has the solutions x = 2 *bc . . .(a = 0, 1,. . ., n+1) if 87Sitzungsber. Bohm. GeselL, Prag, 1897; 1900, No. 30. 88Math. Natur. Berichte aus Ungam, 15, 1897, 41-6. 88ęK. Akad. Wetenschappen te Amsterdam, Verslagen, 5, 1897, 222. 89Nouv. Ann. Math., (3), 17, 1898, 293-5. 90Gottingen Nachrichten, 1900, 184. 91L'intermę5diaire des math., 7, 1900, 253-4; Math. Ann., 54, 1901, 503-4. 92Archiv Math. Phys., (3), 1, 1901, 138-142. 83L'intermddiaire des math., 7, 1900, 254. 84Ueber die Auflosung der Gl. <p(x) =n. . ., Jahres-Bericht Maximilians-Gymn. in Wien, 1900-1, 3-17.