334 History of the Theory of Numbers. [Chap, xi 
that /(p, 0) = 1 , F{p, 0)=0, and any two integers m=np", n='n.p^, where 
the p's are distinct primes and, for any p, a ^ 0, /3 ^ 0. Set 
rP{7n)=Uf{p,a), $=2F(p, a). 
By the usual proof that mn equals the product of the g. c. d. M of m and n 
by their 1. c. m. n, we get 
yPim)4/{n)=^P{M)xl/(jx), <I>(w)+$(n) =$(M)+$(iu). 
In particular, if m and n are relatively prime, 
yl/{m)\p{n) =\p{vin), $(w)+<l>(n) =<l>(mn). 
These hold if i/' is Euler's 0-function, the sum o-(m) of the divisors of 
m or the number T{m) of divisors of ?n ; also, if ^{m) is the number of prime 
factors of vi or the sum of the exponents a in m = Iip''. 
K. Hensel^ proved that the g. c. d. of all numbers represented by a 
polynomial F{u) of degree n with integral coefficients equals the g. c. d. of 
the values of F{u) for any n+1 consecutive arguments. For a polynomial 
of degree ni in Ui, 712 in ^2, • ■ • we have only to use ni + 1 consecutive 
values of ui, 712+ 1 consecutive values of U2, etc. 
F. Klein"*^ discussed geometrically Euclid's g. c. d. process. 
F. ^Vlertens^^ calls a set of numbers primitive if their g. c. d. is unity. 
If 7719^0, k>\, and ai,. . ., o^, m is a primitive set, we can find integers 
Xi,. . ., Xk so that ai-\-mxx,. . ., ak+mxk is a primitive set. Let d be the 
g. c. d. of fli, . . ., Oi- and find 5, ji so that db-\-vi^ = \. Take integral solu- 
tions a of OittiH-. . .+akak = d and primitive solutions ^i not all zero of 
aij3i+ . . . +aii3/; = 0. Then 7i=/3,+6a,('i = l,. . ., k) is a primitive set. 
Determine integers ^ so that 71^1+. . .+7*^^ = 1 and set a:,=/i^<. Then 
Ci+TTix, form a primitive set. 
R. Dedekind^^ employed the g. c. d. d oi a,h, c; the g. c. d. (6, c) =Oi, 
(c, a) = 61, (a, h) = Ci. Then a' = ajd, h' = hi/d, c' = C]/d are relatively prime 
in pairs. Then cf6'c' is the 1. c. m. of 61, Ci, and hence is a divisor of a. Thus 
a = dh'c'a", h = dc'a'b", c = da'h'c". The 7 numbers a', . . .,a" ,. . .,d are called 
the " Kerne" of a, h, c. The generalization from 3 to n numbers is given. 
E. Borel'*^ considered the highest power of a prime p which di\'ides a 
polynomial P{x, y,. . .) with integral coefficients for all integral values of 
X, y,. . .. If each exponent is less than p, we have only to find the highest 
power of p dividing all the coefficients. In the contrary case, reduce all 
exponents below p by use of x^ = x-\-pxi,Xi' = Xi -\-px2,. . . and proceed as 
above with the new polynomial in x, Xi, X2,...,y,yi,.... Then to find all 
arithmetical divisors of a polynomial P, take as p in turn each prime less 
than the highest exponent appearing in P. 
L. Kronecker^^ found the number of pairs of integers i, k having t as 
their g. c. d., where l^i^m, l^k^n. The quotient of this number by 
«Jour. fur Math., 116, 1896, 350-6. 
"Ausgewahlte Kapitel der Zahlentheorie, I, 1896. 
*«Sitzung8berichte Ak. Wiss. Wien (Math.), 106, 1897, II a, 132-3. 
*^Ueber Zerlegungen von Zahlen durch d. grossten gemeinsamen Teller, Braunschweig, 1897. 
"BuU. Sc. Math. Astr., (2), 24 I, 1900, 75-80. Cf. Borel and Drach'^ of Ch. III. 
"Vorlesungen uber Zahlentheorie, I, 1901, 306-312. 
i 
