Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/1

On the expression of a number in the form $\scriptstyle {ax^{2}+by^{2}+cz^{2}+du^{2}}$.

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 $\scriptstyle {a}$, $\scriptstyle {b}$, $\scriptstyle {c}$, $\scriptstyle {d}$ can all positive integers be expressed in the form

   $\scriptstyle {ax^{2}+by^{2}+cz^{2}+du^{2}}$?

   (1·1)

   
   

   I prove in this paper that there are only $\scriptstyle {55}$ sets of values of $\scriptstyle {a}$, $\scriptstyle {b}$, $\scriptstyle {c}$, $\scriptstyle {d}$ for which this is true.

   The more general problem of finding all sets of values of $\scriptstyle {a}$, $\scriptstyle {b}$, $\scriptstyle {c}$, $\scriptstyle {d}$ for which all integers with a finite number of exceptions can be expressed in the form (1·1), 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 (1·1) has one of the special forms

   $\scriptstyle {a(x^{2}+y^{2}+z^{2})+bu^{2}}$

   (1·2),

   
   

   and

   $\scriptstyle {a(x^{2}+y^{2})+b(z^{2}+u^{2})}$

   (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

   $\scriptstyle {a\leq b\leq c\leq d}$

   (2·1).

   
   

   If $\scriptstyle {a>1}$, then $\scriptstyle {1}$ cannot be expressed in the form (1·1); and so

   $\scriptstyle {a=1}$

   (2·2).

   
   

   If $\scriptstyle {b>2}$, then $\scriptstyle {2}$ is an exception; and so

   $\scriptstyle {1\leq b\leq 2}$

   (2·3).

   
   

   We have therefore only to consider the two cases in which (1·1) has one or other of the forms

   $\scriptstyle {x^{2}+y^{2}+cz^{2}+du^{2},\qquad x^{2}+2y^{2}+cz^{2}+du^{2}.}$

   In the first case, if $\scriptstyle {c>3}$, then $\scriptstyle {3}$ is an exception; and so

   $\scriptstyle {1\leq c\leq 3}$

   (2·31).

   
   

   In the second case, if $\scriptstyle {c>5}$, then $\scriptstyle {5}$ is an exception; and so

   $\scriptstyle {2\leq c\leq 5}$

   (2·32).

   
   

   We can now distinguish $\scriptstyle {7}$ possible cases.

   (2·41) $\scriptstyle {x^{2}+y^{2}+z^{2}+du^{2}}$.

   If $\scriptstyle {d>7}$, $\scriptstyle {7}$ is an exception; and so

   $\scriptstyle {1\leq d\leq 7}$

   (2·411).

   
   

   (2·42) $\scriptstyle {x^{2}+y^{2}+2z^{2}+du^{2}}$.