334 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xi that/(p,0) = l, F(p, 0)=0, and any two integers m=IIpa, n=TlpP, where the p's are distinct primes and, for any p, a^O, #^0. Set lKm)=II/(p,a), *«;SF(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. /*, we get In particular, if w and n are relatively prime, \f/ (m) \l/(ri)=\l/ (mn) , $ (m) These hold if ^ is Euler's ^-function, the sum <r(m) of the divisors of m or the number r(m) of divisors of m; also, if <£(m) is the number of prime factors of m or the sum of the exponents a in m=ILpa. K. Hensel44 proved that the g. c. d. of all numbers represented by a polynomial F(u) of degree n with integral coefficients equals Jthe g. c. d. of the values of F(u) for any n+1 consecutive arguments. For a polynomial of degree n^ in ui, n2 in u2, ... we have only to use n^+I consecutive values of uit n2+l consecutive values of u2l etc. F. Klein45 discussed geometrically Euclid's g. c. d. process. F. Mertens46 calls a set of numbers primitive if their g. c. d. is unity. If TWr^O, fc>l, and «i,. . ., ak) m is a primitive set, we can find integers Xi, . . ., xk so that di+mxi, . . ., ak-\-mxk is a primitive set. Let d be the g. c. d. of a!,..., ak and find 5, M so that d8+-mfj,~l. Take integral solutions a of Oiai+. . . +0*0* = ^ and primitive solutions ft not all zero of 0ift+ • • ••H&jkftb = 0. Then 7,-=ft+5oti(i = l, . . ., fc) is a primitive set. Determine integers £ so that 7i£i+. . .+7ifcffc:=l and set x,-= /*£»•• Then ai+mXi form a primitive set. H. Dedekind47 employed the g. c. d. d of a, 6, c; the g. c. d. (6, 0)=^, (c; o) = 61, (a, 6) = Ci. Then a' = ai/d, 6' = &i/d, c' = Cj/d are relatively prime hi pairs. Then db'c' is the 1. c. m. of bi} cb and hence is a divisor of a. Thus a = db'c'a", b = de'a'b", c = da'b'c". The 7 numbers a ', . . . , a7', . . . , d are called the "Kerne" of a, 6, c. The generalization from 3 to n numbers is given. E. Borel48 considered the highest power of a prime p which divides a polynomial P(xt 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 xp = x+pxl,x1p=^x1 +px2). . . and proceed as above with the new polynomial in x, xly x2, . . . , y, y^ . , . . 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. Kronecker49 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, fiir Math., 116, 1896, 350-6. *6Ausgewahlte Kapitel der Zahlentheorie, I, 1896. "Sitzungsberichte Ak. Wiss. Wien (Math.), 106, 1897, II a, 132-3. 47Ueber Zerlegungen von Zahlen durcb d. groaaten gemeinsamen Teller, Braunschweig, 1897. "Bull. Sc. Math. Astr., (2), 24 I, 1900, 75-80. Cf. Borel and Drach180 of Ch. III. «9Vorlesungen iiber Zahlentheorie, I, 1901, 306-312.