124 History of the Theory of Numbers. [Chap, v 
P. Mansion^^ proved (6), showing that </>(m, k) equals <f){m) or 0, accord- 
ing as m is or is not a divisor of k. [Cf. Bachmann, Niedere Zahlentheorie, 
I, 1902, 97-8.] He repeated his*^ ''generalization." He stated that if a 
and b are relatively prime, the products of the 0(a) numbers <a and prime 
to a by the numbers <b and prime to b give the numbers <ab and prime 
to ab [false for a =4, 6 = 3; cf. Mansion^]. His proof of (4) should have 
been credited to Catalan." 
E. Catalan^^ gave a condensation and shght modification of Mansion's*' 
paper. C. Le Paige {ibid., pp. 176-8) proved ]\Iansion's^ theorem that 
every product equals a determinant formed from the factors. 
P. IMansion"" proved that the determinant |cy| of order n equals rriX2- • x^ 
if Cij=Xxp, where p ranges over the divisors of the g. c. d. of i, j. To obtain 
a "generahzation" of Smith's theorem, set Zi = Xi, Z2 = Xi-{-X2,. . ., Zi=J>Xi, 
where d ranges over all the divisors of i. Solving, we get 
where the Vs are defined above.^^ Thus each Cy is a z. For example, if n = 4, - 
21 
21 
2l 
2l 
Zi 
22 
2l 
22 
Zl 
2l 
23 
2i 
21 
22 
2l 
24 
Cii = 
For Zi = i, Xi becomes 4>{i) and we get (6). [As explained in connection with 
Mansion's*^ first paper, the generaUzation is due to Smith.] 
J. J. Sylvester^^ called (/)(7i) the totient T{n) of n, and defined the totitives 
of n to be the integers < n and prime to n. 
F. de Rocquigny^^ stated that, if ^"(A^) denotes <l)\(i>{N)\ , etc., 
if A^ is a prime and m>2, p>2. He stated incorrectly (ibid., 50, 1879, 604) 
that the number of integers ^ P which are prime to N = a^b^ ... is P(l — 1/a) 
(1-1/6).... 
A. Minine*^ noted that the last result is correct for the case in which 
P is divisible by each prime factor a, b,. . . oi N. He wrote symbolically 
nE— for [n/x], the greatest integer ^n/x. By deleting from 1, . . ., P the 
[P/a] numbers di\'isible by a, then the multiples of 6, etc., we obtain for 
the number of integers ^ P which are prime to N the expression 
[equivalent to (5)]. If N, N', N", ... are relatively prime by twos, 
♦^Annalea de la Soc. Sc, Bru.\elles, 2, II, 1877-8, 211-224. Reprinted in Mansion's Sur la 
th^orie des nombres, Gand, 1878, §3, pp. 3-16. 
"Nouv. Corresp. Math., 4, 1878, 103-112. 
«Bull. Acad. R. Sc. de Belgique, (2), 46, 1878, 892-9. 
«Amer. Jour. Math., 2, 1879, 361, 378; Coll. Papers, 3, 321, 337. Nature, 37, 1888, 152-3. 
«Le8 Mondes, Revue Hebdom. des Sciences, 48, 1879, 327. 
"Ibid., 51, 1880, 333. Math. Soc. of Moscow, 1880. Jour, de math. 616in. et sp^c, 1880, 278. 
