Mr Rainanujau, On t/ie ej^pressiun of a nv/inber, etc. 11 



On the expression of a number in the form ax- + by- + cz- + dv-. 

 By S. Ramanujan, B.A., Trinity College. (Communicated by 

 Mr G. H. Hardy.) 



[Received 19 September 1916 ; read October 30, 1916.] 



1. It is well known that all positive integers can be expressed 

 as the sum of four squares. This naturally suggests the question : 

 For what positive integral values of a, b, c, d can all positive 

 integers be expressed in the form 



ax- + by- + cz- + du" ? (ll) 



I prove in this paper that there are only 55 sets of values of 

 a, b, c, d for which this is true. 



The more general problem of finding all sets of values of 

 a, b, c, d, for which all integers luith a finite number of exceptions 

 can be expressed in the form (II), is much more difficult and 

 interesting. I have considered only very special cases of this 

 problem, with two variables instead of four ; namely, the cases in 

 which (I'l) has one of the special forms 



a{x^ + y'- + z^) + bu- (1-2), 



and a{x'^ +y-)-\-b{z''-\-u?) (1-3). 



These two cases are comparatively easy to discuss. In this 

 paper I give the discussion of (1"2) only, reserving that of (1"3) 

 for another paper. 



2. Let us begin with the first problem. We can suppose, 

 without loss of generality, that 



a^b^c^d (2'1). 



If a > 1, then 1 cannot be expressed in the form (I'l) ; and so 



a = l (2-2). 



If b> 2, then 2 is an exception ; and so 



1<6^2 (2-3). 



We have therefore only to consider the two cases in which (1"1) 

 has one or other of the forms i 



X- + y- + cz^ + du-, X- + 2y'- + cz- + dii-. 

 In the first case, if c> 3, then 3 is an exception ; and so 



l^c^3 (2-31). 



In the second case, if c > 5, then 5 is an exception ; and so 



2^c$5 (2-32). 



We can now distinguish 7 possible cases. 



(2-41) x"- -\- y- + z'- + du?. 

 If rf > 7, 7 is an exception ; and so 



X^d^l (2-411). 



(2-42) X- + y- + 2z- + du'. 



