Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/2

 12 If $$\scriptstyle{d>14}$$, $$\scriptstyle{14}$$ is an exception; and so (2·43) $\scriptstyle{x^2+y^2+3z^2+du^2}$.

If $$\scriptstyle{d>6}$$, $$\scriptstyle{6}$$ is an exception; and so (2·44) $\scriptstyle{x^2+2y^2+2z^2+du^2}$.

If $$\scriptstyle{d>7}$$, $$\scriptstyle{7}$$ is an exception; and so (2·45) $\scriptstyle{x^2+2y^2+3z^2+du^2}$.

If $$\scriptstyle{d>10}$$, $$\scriptstyle{10}$$ is an exception; and so (2·46) $\scriptstyle{x^2+2y^2+4z^2+du^2}$.

If $$\scriptstyle{d>14}$$, $$\scriptstyle{14}$$ is an exception; and so (2·47) $\scriptstyle{x^2+2y^2+5z^2+du^2}$.

If $$\scriptstyle{d>10}$$, $$\scriptstyle{10}$$ is an exception; and so

We have thus eliminated all possible sets of values of $$\scriptstyle{a}$$, $$\scriptstyle{b}$$, $$\scriptstyle{c}$$, $$\scriptstyle{d}$$, except the following 55: