Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/3

 Rh Of these 55 forms, the 12 forms have been already considered by Liouville and Pepin.

I shall now prove that all integers can be expressed in each of the 55 forms. In order to prove this we shall consider the seven cases (2·41)—(2·47) of the previous section separately. We shall require the following results concerning ternary quadratic arithmetical forms.

The necessary and sufficient condition that a number cannot be expressed in the form is that it should be of the form

Similarly the necessary and sufficient conditions that a number cannot be expressed in the forms

are that it should be of the forms