16 Mr lia/uanujan, On tJie expression of a nwniber 



or is of any of the forms 



4yt+2, 4yb + 3, 8^•^-o, 16A; + 12, 32/.^ + 20 ...(5-22), 

 then all integers save a finite number, and in fact all integers from 

 4?i onwards at any rate, can be expressed in the form (5*2) ; but 

 that for the remaining values of n there is an infinity of integers 

 which cannot be expressed in the form required. 



In proving the first result we need obviously only consider 

 numbers of the form 4*^ (8yu, + 7) greater than n, since otherwise 

 we may take w = 0. The numbers of this form less than n are 

 plainly among the exceptions. 



6. I shall consider the various cases which may arise in 

 order of simplicity. 



(6-1) 7^ = (mod 8). 

 There are an infinity of exceptions. For suppose that 

 N = Sfju + 1. 

 Then the number 



N - nu- = 7 (mod 8) 

 cannot be expressed in the form cc- + y- + z'^. 



(6-2) n=2 (mod 4). 



There is only a finite number of exceptions. In proving this 

 we may suppose that iV=4^(8/i + 7). Take u=l. Then the 

 number 



]SF - oiu^ = 4'" (S/M + 1) - n 

 is congruent to 1, 2, 5, or 6 to modulus 8, and so can be expressed 

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



Hence the only numbers which camiot be expressed in the 

 form (5"2) in this case are the numbers of the form 4^(8/i+ 7) not 

 exceeding n. 



(6-3) n=h (mod 8). 



There is only a finite number of exceptions. We may suppose 

 again that i\r = 4^ (8/i + 7). First, let X =|= 1 . Take u=\. Then 



N - nu- = 4^ (8/i + 7) - n = 2 or 3 (mod 8). 

 If X = 1 we cannot take u = l, since 



N - n = 7 (mod 8) ; 

 so we take u = 2. Then 



JSf- nu- = V (8/i + 7) - 4n = 8 (mod 32). 



In either of these cases N — nu^ is of the form cc'^ -\-y'^ + z". 



Hence the only numbers which cannot be expressed in the 

 form (5*2) are those of the form 4^ (8/a + 7) not exceeding ??, and 

 those of the form 4 (8/ti + 7) lying between n and 4?l 



