in the fornt (i.r- + bij'^ -\- cz- + (hr ID 



In the first place n must be odd ; otherwise the odd iiuinbers 

 cannot be expressed in this form. Suppose then that n is odd. 

 I shall show that all integers save a finite number can be expressed 

 in the form (7"1): and that the numbers which cannot be so 

 expressed are 



(i) the odd numbers less than n, 



(ii) the numbers of the form 4^^ (16yu. + 14) less than 4n, 

 (iii) the numbers of the form n-\- 4^(16ya+ 14) greater than 



n and less than 9w, 

 (iv) the numbers of the form 



cn-\-¥{l(w+U\ (i^ = 0, 1, 2, 3, ...), 



greater than 9/i and less than 25w, where c = 1 if 

 n = \ (mod 4), c = 9 if w = 3 (mod 4), « = 2 if n^= 1 

 (mod 16); and /c > 2 if /(-=9 (mod 16). 



First, let us suppose N even. Then, since n is odd and N is 

 even, it is clear that u must be even. Suppose then that 



We have to show that M can be expressed in the form 



x"->r y- -\- Z' + 27?,?'- (7-2). 



Since %i = 2 (mod 4), it follows from (6'2) that all integers except 

 those which are less than 2n and of the form 4-^ (8//. + 7) can be 

 expressed in the form (7*2). Hence the only even integers which 

 cannot be expressed in the form (7'1) are those of the form 

 4^(16/* + 14) less than 4n. 



This completes the discussion of the case in which N is even. 

 If N is odd the discussion is more difficult. In the first place, 

 all odd numbers less than n are plainl37^ among the exceptions. 

 Secondly, since n and N are both odd, u must also be odd. We 

 can therefore suppose that 



iV = ?i + 2il/, xv" = 1 + 8A, 



where A is an integer of the form |^•(^• + l), so that A may 

 assume the values 0, 1, 3, 6, .... And we have to consider 

 whether n + 2if can be expressed in the form 



2 0r2 + 2/'^ + ^'^) + w(l +8A), 

 or M in the form 



«- + 2/- + ^- + 4nA (7-3). 



If M is not of the form 4^ (8/a + 7), we can take A = 0. If it is 

 of this form, and less than 4?i, it is plainly an exception. These 

 numbers give rise to the exceptions specified in (iii) of section 7. 

 We may therefore suppose that M is of the form 4^ (8^* + 7) and 

 greater than 4/?. 



2—2 



