# Page:On the expression of a number in the form ππ₯Β²+ππ¦Β²+ππ§Β²+ππ’Β².djvu/3

Of these 55 forms, the 12 forms

 1, 1, 1, 2 1, 1, 2, 4 1, 2, 4, 8 1, 1, 2, 2 1, 2, 2, 4 1, 1, 3, 3 1, 2, 2, 2 1, 2, 4, 4 1, 2, 3, 6 1, 1, 1, 4 1, 1, 2, 8 1, 2, 5, 10
have been already considered by Liouville and Pepin[1].

## 3.

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 (2Β·41)β(2Β·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

 $x^2+y^2+z^2$ (3Β·1)
is that it should be of the form


 $4^\lambda(8\mu+7),\quad(\lambda=0,1,2\ \ldots,\mu=0,1,2,\ldots)$ (3Β·11).

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

 $x^2+\ y^2+2z^2$ (3Β·2),



 $x^2+\ y^2+3z^2$ (3Β·3),



 $x^2+2y^2+2z^2$ (3Β·4),



 $x^2+2y^2+3z^2$ (3Β·5),



 $x^2+2y^2+4z^2$ (3Β·6),



 $x^2+2y^2+5z^2$ (3Β·7),
are that it should be of the forms


 $4^\lambda(16\mu+14)$ (3Β·21),



 $9^\lambda(\ 9\mu+\ 6)$ (3Β·31),



 $4^\lambda(\ 8\mu+\ 7)$ (3Β·41),



 $4^\lambda(16\mu+10)$ (3Β·51),



 $4^\lambda(16\mu+14)$ (3Β·61),
 $25^\lambda(25\mu+10)$ or $25^\lambda(25\mu+15)$[2] (3Β·71).
1. β There are a large number of short notes by Liouville in vols. vβviii of the second series of his journal. See also Pepin, ibid., ser. 4, vol vi, pp. 1β67. The object of the work of Liouville and Pepin is rather different from mine, viz. to determine, in a number of special cases, explicit formulae for the number of representations, in terms of other arithmetical functions.
2. β Results (3Β·11)β(3Β·71) may tempt us to suppose that there are similar simple results for the form $ax^2+by^2+cz^2$, whatever are the values of $a$, $b$, $c$. It appears, however, that in most cases there are no such simple results. For instance,