in the form (ur- + hy" + cz" + du? 15 



If ya^l, take« = 2^+\ Then 



i\r-rf«;^ = 4^(8/^-5). 



In neither of these cases is TV — d}f- of the form 4^ (8/i + 7), 

 and therefore in either case it can be expressed in the form 



X- + 2/2 4- z". 



Finally, let d=l. If ^u, is equal to 0, 1, or 2, take « = 2\ 

 Then N - d^ir is equal to 0, 2 . 4^+\ or 4^+-. If /a^S, take 

 M. = 2^+1. Then 



iV-*/,^=4^(8/x-21). 



Therefore in either case N — du- can be expressed in the form 



a;2 + 2/2 4- Z-. 



Thus in all cases N is expressible in the form (4'1). Similarly 

 we can dispose of the remaining cases, with the help of the results 

 stated in § 3. Thus in discussing (2-42) we use the theorem that 

 every number not of the form (3'21) can be expressed in the form 

 (3*2). The proofs differ only in detail, and it is not worth while 

 to state them at length. 



5. We have seen that all integers without any exception can 

 be expressed in the form 



m. {x^ +'if + z^) + mi'^ (5-1), 



when m = \, li^n^l, 



and m= 2, n = 1. 



We shall now consider the values of m and n for which all 

 integers with a finite number of exceptions can be expressed in 

 the form (5'1), 



In the first place 7?? must be 1 or 2. For, if m > 2, we can 

 choose an integer v so that 



7iu' ^ V (mod 7)1) 

 for all values of u. Then 



(nifx + v) — mi^ 

 m 



where fi is any positive integer, is not an integer ; and so 7nfj, + v 

 can certainly not be expressed in the form (5'1). 



We have therefore only to consider the two cases in which m is 

 1 or 2. First let us consider the form 



cc- + 2/- + z- + nit^ (5'2). 



I shall show that, when n has any of the values 



1, 4, 9, 17, 25, 36, 68, 100 (5-21), 



