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

 Rh On the expression of a number in the form $\scriptstyle{ax^2+by^2+cz^2+du^2}$. By, B.A., Trinity College. (Communicated by Mr .) [Received 19 September 1916; read October 30, 1916.]

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

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

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. Let us begin with the first problem. We can suppose, without loss of generality, that If $$\scriptstyle{a>1}$$, then $$\scriptstyle{1}$$ cannot be expressed in the form (1·1); and so If $$\scriptstyle{b>2}$$, then $$\scriptstyle{2}$$ is an exception; and so 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 In the second case, if $$\scriptstyle{c>5}$$, then $$\scriptstyle{5}$$ is an exception; and so

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