18 31r Ramanujan, On the ex'pression of a number 



where k = 14 . V-''-- - nv - ^ {n + 7) 



is an integer ; and so N — nu- is not of the form x- +y'^ + z\ 

 In order to prove (ii) we may suppose, as usual, that 



N = 4^ (Sfi + 7). 

 IfX = 0, take w=l. Then 



iV - nil'' = 8// + 7 - 7? = 6 (mod 8). 

 If X^l, take w= 2^-1. Then 



where k = 4< (fji + l) -^{n + 7). 



In either case the proof may be completed as before. Thus the 

 only numbers which cannot be expressed in the form (5'2), in 

 this case, are those of the form 8/u. + 7 not exceeding n. In 

 other words, there is no exception when n = 1 ; 7 is the only 

 exception when n = 9; 7 and 15 are the only exceptions when 

 n = 17 ; 7, 15 and 23 are the only exceptions when n = 25. 



(6-Q) n = 4 (mod 32). 

 By arguments similar to those used in (6'5), Ave can show that 

 (i) if w ^ 132, there is an infinity of integers which cannot 



be expressed in the form (5*2) ; 

 (ii) if n is equal to 4, 36, 68, or 100, there is only a finite 

 number of exceptions, namely the numbers of the 

 form 4'^ (8yu, + 7) not exceeding n. 



(6-7) ?i = 20 (mod 32). 



By arguments similar to those used in (6'3), we can show that 

 the only numbers which cannot be expressed in the form (5'2) are 

 those of the form 4^ (8yLi +7) not exceeding n, and those of the form 

 4^(8/A + 7) lying between n and 4n. 



(6-8) n= 12 (mod 16). 



By arguments similar to those used in (6'4), we can show that 

 the only numbers which cannot be expressed in the form (5"2) are 

 those of the form 4^ (Sfi + 7) less than n, and those of the form 



n + 4>^(8v + 1), (i/ = 0, 1, 2, 3, ...), 

 lying between n and 4>i, where /c = 2 if n is of the form 4 (8k + 3) 

 and AC > 2 if w is of the form 4 (8A; +7). 



We have thus completed the discussion of the form (5 2), and 

 determined the exceptional values of iV precisely whenever they 

 are finite in number. 



7. We shall proceed to consider the form 



2 (ic^ +y" + z') + mir .(7-1). 



