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

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13
in the form $\scriptstyle{ax^2+by^2+cz^2+du^2}$
1. 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].
2. 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
 $\scriptstyle{x^2+y^2+z^2}$ (3Β·1)

is that it should be of the form

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

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

 $\scriptstyle{x^2+\ y^2+2z^2}$ (3Β·2),
 $\scriptstyle{x^2+\ y^2+3z^2}$ (3Β·3),
 $\scriptstyle{x^2+2y^2+2z^2}$ (3Β·4),
 $\scriptstyle{x^2+2y^2+3z^2}$ (3Β·5),
 $\scriptstyle{x^2+2y^2+4z^2}$ (3Β·6),
 $\scriptstyle{x^2+2y^2+5z^2}$ (3Β·7),

are that it should be of the forms

 $\scriptstyle{4^\lambda(16\mu+14)}$ (3Β·21),
 $\scriptstyle{9^\lambda(\ 9\mu+\ 6)}$ (3Β·31),
 $\scriptstyle{4^\lambda(\ 8\mu+\ 7)}$ (3Β·41),
 $\scriptstyle{4^\lambda(16\mu+10)}$ (3Β·51),
 $\scriptstyle{4^\lambda(16\mu+14)}$ (3Β·61),
 $\scriptstyle{25^\lambda(25\mu+10)}$ or $\scriptstyle{25^\lambda(25\mu+15)}$[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 $\scriptstyle{ax^2+by^2+cz^2}$, whatever are the values of $\scriptstyle{a}$, $\scriptstyle{b}$, $\scriptstyle{c}$. It appears, however, that in most cases there are no such simple results. For instance,