14 Mr Raiitavujan, On tJie exjyression. of a nninher 



The result concerning ./- + y^ + z- is due to Cauchy : for a proof 

 see Landau, Handhuch der LeJtre von der Verteilung der Prim- 

 zahlen, p. 550. The other results can be proved in an analogous 

 manner. The form x- + y" + ^z"- has been considered by Lebesgue, 

 and the form x'^-\-y'^-{-'^Z' by Dirichlet. For references see Bach- 

 mann, Zahlentheorie, vol. iv, p. 149. 



4. We proceed to consider the seven cases (2'41) — (2*47). In 

 the first case we have to show that any number N can be expressed 

 in the form 



N' = x- + y- + z- + du" (4- 1 ), 



d being any integer between 1 and 7 inclusive. 



If JSf is not of the form 4^(8yLt + 7), we can satisfy (4-1) with 

 u = 0. We may therefore suppose that iV^= 4^ (S/jl + 7). 



First, sup]3ose that d has one of the values 1, 2, 4, 5, 6. 

 Take u = 2\ Then the number 



N-du' = ^^(8fM+7-d) 



is plainly not of the form 4^(8/4 + 7), and is therefore expressible 

 in .the form x^ + y^ + z^. 



Next, let d = S. If /i = 0, take u = 2\ Then 



N - dii- = 4^+1. 



the numbers which are not of the form .r- + 2?/- + 10^- are those belonging to one 

 or other of the four classes 



25^(8^ + 7), 25^(25^ + 5), 25^ (25/x + 15) , 25^ (25/^+20). 

 Here some of the numbers of the first class belong also to one of the next three 

 classes. 



Again, the even numbers which are not of the form x'^ + ij- + lOz- ai'e the numbers 



4^(16^ + 6), 

 while the odd numbers that are not of that form, viz. 



3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391, ... 

 do not seem to obey any simple law. 



I have succeeded in finding a law in the following six simple cases: 

 •«■''+ ?/2 + 4,--, 

 X-+ y- + 5z-, 

 x^+ y' + 6z', 

 x^+ y^ + 8z-, 

 x^ + 2y-^ + Qz\ 

 .r2 + 2y- + 8^2. 

 The numbers which are not of these forms are the numbers 

 4^(8^ + 7) or (8^1 + 3), 

 4^(8^ + 3), 

 9^(9ya + 3), 

 4^(16/x+14), (16m + 6), or (4^ + 3), 

 4^ (8m +5), 

 4^^(8^4-7) or (8^ + 5). 



