282 Lanavicensis on a Theorem in Numbers. 



The number of simple products which coalesce to form such 

 a term will evidently be 



n(5) 



n(«)n(^)n(7)n(8)n(e)' 



and such number will therefore be of the form 5 m, except for the 

 case when « = 0, /3 = 0, 7 = 5, S=0, e = 0, for which case the 

 number in question is unity. 



Now every one of these 5m products will have for its coeffi- 

 cient some power of p. Let k of them be of the form p''^ ; then 

 the sum of the coefficients for these k products is k. As regards 

 the 5m — k remaining products, since their sum remains the 

 same whether we write for p, \ X~, 7<P, or X'^ (X being any ])rime 

 5th root of unity), it is clear that they will consist of repetitions 

 of a sum of products each equivalent in value to p + p'^ + p^ + p^. 

 Hence we shall have 



5m — k = {5 — l)i, 



and the coefficient of A".B^C\D^F will be k-i or 5m-5i, 

 that is to say, Avill contain 5 (the p of the example before us). 

 Hence, then, C, the coefficient of x'^ in the transformed equation, 

 will be of the form C''' + 5M. And since C^ — C by Fermat's 

 theorem contains 5, C' — C, which is (C^ — C + 5M), will also eon- 

 tain 5. And the like will be true for every other conjugate pair 

 of coefficients of the two allied functions in our example, or in 

 any other that maybe taken, which proves the proposition that 

 was to be established. 

 September 1, 1859. 



P.S. I subjoin a proof of another theorem alluded to, but left 

 without demonstration, in M. Lebesguc's valuable pa])er. 



Let (fjn^' mean the primitive factor of a?" — ! («. e, tlie function 

 which contains all its prime factors). If n is of the form a« 



xa^ — l 

 (a being a prime number), ^a'^— — ^;zri — ^j and obviously 



becomes a when a; = l. The theorem in question affirms that 

 if n is of the form ««, 0„(1) = «, and if n is of any other form, 



Suppose the theorem proved for n = aP.h''. &' . . .V when 

 r) + q + r+...+t<s. 



Nowlet n = rt«.A'^.c>'...A, and a + /3-F7. . . +X=s. Then 



_ .Z'" — 1 



where v, v', v" ... represent all the possible values of a"-' . h^' . c^' ... l\ 

 with the restriction that a, /S', y'...X' are limited not to exceed 



