20 Mr Ramamtjan, On the eo;pression of a number 



8. In order to complete the discussion, we must consider 

 the three cases in which n = 1 (mod 8), n = 5 (mod 8), and 

 n = S (mod 4) separately. 



(8-1) /I- 1 (mod 8). 

 If X is equal to 0, 1, or 2, take A = 1. Then 

 M - 4?iA = 4^ (8/i + 7) - 4?i 

 is of one of the forms 



8z; + 3, 4 (Sv + 3), 4 {8v + 6). 



If A. ^ 3 we cannot take A = 1, since if — 4?iA assumes the 

 form 4 (8i/ + 7) ; so we take A = 3. Then 



M - 4n A = 4^ (SyLt + 7) - 12n 



is of the form 4 {8v + 5). In either of these cases M — 4nA is of 

 the form x^ + y'^ -\- z^. Hence the only values of M, other than 

 those already specified, which cannot be expressed in the form 

 (7*3). are those of the form 



4«(8i; + 7), (z/ = 0, 1,2, ...,«>2), 



lying between 4?i and 12??. In other words, the only numbers 

 greater than 9'n which cannot be expressed in the form (71), in 

 this case, are the numbers of the form 



n+4«(8j; + 7), (i^ = 0, 1, 2, ..., /c> 2), 



lying between 9?i and 25?l 



(8-2) ?i = 5 (mod 8). 

 If X 4= 2, take A =1. Then 



ili - 4wA = 4^ (8/i + 7) - 4?i 

 is of one of the forms 



8i/ + 3, 4 (8z/ + 2), 4 (8z/ + 3). 



If \ = 2, we cannot take A = l, since ilf— 4?iA assumes the 

 form 4 (8v + 7) ; so we take A = 3. Then 



M- 4mA = 4^ (8/A + 7) - \%i 



is of the form 4 (8y + 5). In either of these cases M — 4/? A is of 

 the form (x? ■\-y'^ A- z^. Hence the only values of M, other than 

 those already specified, which cannot be expressed in the form 

 (7-3), are those of the form 16 (8^t + 7) lying between 4n and 12?i. 

 In other words, the only numbers greater than 2n which cannot 

 be expressed in the form (7"1), in this case, are the numbers of the 

 form n + 4"^(16/ti + 14) lying between 9?? and 2.5». 



