Chap. XI] GREATEST COMMON DiVISOR. 333 
Q by (x — p — l)... {x — 2p) and call the quotient Q' and remainder R'. 
Then must X^R'+XopQ'^O and hence XpR'=0 (mod p"). Thus if 1^2 is 
the exponent of the highest power of p dividing the coefficients of R', we 
have At^M2 + l- In general, if [j.^ and X^-i are the exponents of the highest 
powers of p dividing the coefficients of the remainder i?'^~^^ and X^k-Dp 
identically, then fi^ iJLk+\k-i' Finally, if l = [m/p], /x^X^. The extension 
to several variables is said to present difficulties. [For simpler methods, 
see Hensel^^ and Borel.^^] It is noted (p. 323) that 
are identically divisible by p". It is conjectured (p. 328) that/(a;)/iV repre- 
sents an infinitude of primes when f{x) is irreducible. 
E. Cesaro" and J. J. Sylvester^^ proved that the probabihty that two 
numbers taken at random from 1 , . . . , n be relatively prime is Q/tt^ asymp- 
totically. 
L. Gegenbauer^^ gave 16 sums involving the g. c. d. of several integers 
and deduced 37 asymptotic theorems such as the fact that the square of 
the g. c. d. of four integers has the mean value IS/tt^. He gave the mean 
of the kth. power of the g. c. d. of r integers. 
J. Neuberg^^" noted that, if two numbers be selected at random from 
1, . . .,N, the probability that their sum is prime to N is k=cf){N) 0Tk/{N—l) 
according as N is odd or even. 
T. J. Stieltjes,^^ starting with a set of n integers, replaced two of them 
by their g. c. d. and 1. c. m., repeated the same operation on the new set, 
etc. Finally, we get a set such that one number of every pair divides the 
other. Such a reduced set is unique. The 1. c. m. of a, . . . , ? can be 
expressed (pp. 14-16) as a product a'. . J' of relatively prime factors divi- 
ding a,...,l, respectively. The 1. c. m. (or g. c. d.) oi a,h,. . .,1 equals the 
quotient oi P = ab. . .Ihy the g. c. d. (or 1. c. m.) of P/a, P/b, . . . , P/l. 
E. Lucas^^ gave theorems on g. c. d. and 1. c. m. 
L. Gegenbauer^^" considered in connection with the theory of primes, 
the g. c. d. of r numbers with specified properties. 
J. Hacks^^ expressed the g. c. d. of m and n in the forms 
ql^]-.n^.^n, 2'|;[f]+2'}:g]-2[|][|]-^. 
where € = or 1 according as m, n are both or not both even. 
J. Hammond^^ considered arbitrary functions / and F oi p and a, such 
"Mathesis, 1, 1881, 184; Johns Hopkins Univ. Circ, 2, 1882-3, 85. 
"Johns Hopkins Univ. Circ, 2, 1883, 45; Comptes Rendus Paris, 96, 1883, 409; Coll. Papers, 3, 
675; 4, 86. 
"Sitzungsberichte Ak. Wiss. Wien (Math.) 92, 1885, II, 1290-1306. 
39«Math. Quest. Educ. Times, 50, 1889, 113-4. 
^''Sur la theorie des nombres, Annales de la fac. des sciences de Toulouse, 4, 1890, final paper. 
^iTheorie des nombres, 1891, 345-6; 369, exs. 4, 5. 
""Monatshefte Math. Phys., 3, 1892, 319-335. 
*2Acta Math., 17, 1893, 208. 
^Messenger Math. 24 1894^5 17-19. 
