Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/4

 14 The result concerning $$\scriptstyle{x^2+y^2+z^2}$$ is due to Cauchy: for a proof see Landau, Handbuch der Lehre von der Verteilung der Primzahlen, p. 550. The other results can be proven in an analogous manner. The form $$\scriptstyle{x^2+y^2+2z^2}$$ has been considered by Lebesgue, and the form $$\scriptstyle{x^2+y^2+3z^2}$$ by Dirichlet. For references see Bachmann, Zahlentheorie, vol. , p. 149. We proceed to consider the seven cases (2·41)—(2·47). In the first case we have to show that any number $$\scriptstyle{N}$$ can be expressed in the form$$\scriptstyle{d}$$ being any integer between $$\scriptstyle{1}$$ and $$\scriptstyle{7}$$ inclusive.

If $$\scriptstyle{N}$$ is not of the form $$\scriptstyle{4^\lambda(8\mu+7)}$$, we can satisfy (4·1) with $$\scriptstyle{u=0}$$. We may therefore suppose that $$\scriptstyle{N=4^\lambda(8\mu+7)}$$.

First, suppose that $$\scriptstyle{d}$$ has one of the values $$\scriptstyle{1}$$, $$\scriptstyle{2}$$, $$\scriptstyle{4}$$, $$\scriptstyle{5}$$, $$\scriptstyle{6}$$. Take $$\scriptstyle{u=2^\lambda}$$. Then the number$\scriptstyle{N-du^2=4^\lambda(8\mu+7-d)}$is plainly not of the form $$\scriptstyle{4^\lambda(8\mu+7)}$$, and is therefore expressible in the form $$\scriptstyle{x^2+y^2+z^2}$$.

Next, let $$\scriptstyle{d=3}$$. If $$\scriptstyle{\mu=0}$$, take $$\scriptstyle{u=2^\lambda}$$. Then

$\scriptstyle{N-du^2=4^{\lambda+1}}$.|undefined