140 History of the Theory of Numbers. [Chap, v 
E. Cahen^^^ gave F. Arndt's^^ proof without reference. 
A. Cunningham^" tabulated all solutions N of 0(iV)=2' for r = 4, 6, 8, 
9, 10, 11, 12, 16, each solution being a product of a power of 2 by distinct 
primes 22"+ 1. 
J. Hanmiond^-^ noted that, if 'Zf{k/n)=F{7i) or <l>(n), according as the 
summation extends over all positive integers k from 1 to n or only over 
such of them as are prime to n, then Z$(d)=F(n). This becomes (4) 
when /is constant. 
R. Ratat^29 ^oted that 0(n) = 0(n + l) for n= 1, 3, 15, 104. For n<125, 
2n7^2, 4, 16, 104, he verified that (/)(2n=t l)>0(2n). 
R. Goormaghtigh^^o ^^^^^ ^j^j^^ 0(^^) = <^(^i_|_l) also for n= 164, 194, 255 
and 495. He gave very special results on the solution of (f){x) = 2a. 
Formulas involving cf) are cited under Lipschitz,'''°' ^^ Cesaro,^^ Ham- 
mond,^" and Knopp^^^ of Ch. X, Hammond^ of Ch. XI, and RogeP« of 
Ch. XVIII. Cunningham^^ of Ch. VII gave the factors of (t>{f). Dede- 
kind^^ of Ch. VIII generalized ^ to a double modulus. Minin^^° of Ch. 
X solved 0(iV)=r(A^). 
Sum 0fc(n) of the A:th Powers of the Integers ^n and Prime to n. 
A. Cauchy^^^ noted that (piin) is divisible by n if n>2, since the integers 
<n and prime to n may be paired so that the sum of the two of any pair is n. 
A. L. Crelle^^ (p. 80, p. 84) noted that (^i(n) = |n<^(n). The proof 
follows from the remark by Cauchy. 
A. Thacker^^*^ defined (f)k{n) and noted that it reduces for k = to Euler's 
<i>{n). Set St(2) = l'"-|-2^+ . . .+2^n = a°6V. . ., where a, 6, . . . are distinct 
primes. By deleting the multiples of a, then the remaining multiples of 
b, etc., he proved that 
Mn)=sM -2a^s.(^) +2 aVs,(^) - S ^a*feVs,(^) + . . . , 
where the summation indices range over the combinations of a, 5, c, . . . one, 
two, ... at a time. In the second paper, he proved Bernoulli's^^"" formula 
where Bi, Bz,... are the Bernoullian numbers. Then, by substitution, 
^^(^)=^n(l-i)+§(J)5,n^-^n(l-a)-i(3)53n*-^n(l-a^) 
i^TWorie des nombres, I, 1914, 393. 
i"Math. Quest. Educ. Times, 27, 1915, 103-6. 
128/bid., 29, 1916, 53. 
i"L'interm(5diaire des math., 24, 1917, 101-2. 
"»/6ui., 25, 1918, 42-4. 
"'M6m. Ac. Sc. de I'Institut de France, 17, 1840, 565; Oeuvres, (1), 3, 272. 
"ojour. fur Math., 40, 1850, 89-92; Cambridge and Dublin Math. Jour., 5, 1850, 243. Repro- 
duced, with errors as to signs, by Zerr, Amer. Math. Monthly, 5, 1898, 93-5. Cf. E. 
Prouhet, Xouv. Ann. Math., 10, 1851, 324-330. 
"""Jacques Bernoulli, Are conjectandi, 1713, 95-7. 
