140 HISTORY OF THE THEORY OF NUMBERS. [CHAP, v E. Cahen126 gave F. Arndt's19 proof without reference. A. Cunningham127 tabulated all solutions N of cf>(N) =2r for r = 4, 6, 8, 9, 10, 11, 12, 16, each solution being a product of a power of 2 by distinct primes 22*+l, J. Hammond128 noted that, if 2f(k/ri)=F(ri) or &(ri)* 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 S$(d)=F(w). This becomes (4) when / is constant. R. Ratat129 noted that 0(n) = «(n+l) for n= 1, 3, 15, 104. For n< 125, 2w^2, 4, 16, 104, he verified that </>(2n±l)>0(2n). R. Goormaghtigh130 noted that 0(n) = 0(n+l) also for n=164, 194, 255 and 495. He gave very special results on the solution of ct>(x) = 2a. Formulas involving 0 are cited under Lipschitz,50' 56 Cesaro,61 Hammond,111 and Knopp160 of Ch. X, Hammond43 of Ch. XI, and Rogel243 of Ch. XVIII. Cunningham95 of Ch. VII gave the factors of 0(pfc). Dede-kind71 of Ch. VIII generalized <t> to a double modulus. Minin120 of Ch. X solved <t>(N)=r(N). SUM <f>k(n) OF THE &TH POWERS OF THE INTEGERS §n AND PRIME TO n. A. Cauchy149 noted that <£i(n) 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. Crelle17 (p. 80, p. 84) noted that ^(n) =Jn0(n). The proof follows from the remark by Cauchy. A. Thacker160 defined <j>k(ri) and noted that it reduces for /b = 0 to Euler's <t>(ri) . Set sh(z) = l*+2*+ . . . -f-z*, n = aa6V . . . , where a, 6, ... are distinct prunes. By deleting the multiples of a, then the remaining multiples of 6, etc., he proved that where the summation indices range over the combinations of a, 6, c, ... one, two, ... at a time. In the second paper, he proved Bernoulli's1500 formula where BI, B3) . . . are the Bernoullian numbers. Then, by substitution, /A mTii6orie des nombres, I, 1914, 393. 127Math. Quest. Educ. Times, 27, 1915, 103-6. l28/6id., 29, 1916, 53. 129L'interm6diaire des math., 24, 1917, 101-2. 13°76td., 25, 1918, 42-4. "9M6m. Ac. Sc. de 1'Institut de France, 17, 1840, 565; Oeuvres, (1), 3, 272. w°Jour. 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, Nouv. Ann. Math., 10, 1851, 324-330. 180aJacques Bernoulli, Ars conjectandi, 1713, 95-7.