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

3. 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. iv, p. 149.
4. 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

   $\scriptstyle {N=x^{2}+y^{2}+z^{2}+du^{2}}$

   (4·1),

   $\scriptstyle {d}$ 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

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

the numbers which are not of the form $\scriptstyle {x^{2}+2y^{2}+10z^{2}}$ are those belonging to one or other of the four classes


$\scriptstyle {25^{\lambda }(8\mu +7),\quad 25^{\lambda }(25\mu +5),\quad 25^{\lambda }(25\mu +15),\quad 25^{\lambda }(25\mu +20).}$




Here some of the numbers of the first class belong also to one of the next three classes.


Again, the even numbers which are not of the form $\scriptstyle {x^{2}+y^{2}+10z^{2}}$ are the numbers


$\scriptstyle {4^{\lambda }(16\mu +6)}$,


while the odd numbers that are not of that form, viz.


$\scriptstyle {3,~7,~21,~31,~33,~43,~67,~79,~87,~133,~217,~219,~223,~253,~307,~391,~\ldots }$




do not seem to obey any simple law.


I have succeeded in finding a law in the following six simple cases:


${\begin{array}{rcrcr}\scriptstyle {x^{2}}&\scriptstyle {+}&\scriptstyle {y^{2}}&\scriptstyle {+}&\scriptstyle {4z^{2},}\\\scriptstyle {x^{2}}&\scriptstyle {+}&\scriptstyle {y^{2}}&\scriptstyle {+}&\scriptstyle {5z^{2},}\\\scriptstyle {x^{2}}&\scriptstyle {+}&\scriptstyle {y^{2}}&\scriptstyle {+}&\scriptstyle {6z^{2},}\\\scriptstyle {x^{2}}&\scriptstyle {+}&\scriptstyle {y^{2}}&\scriptstyle {+}&\scriptstyle {8z^{2},}\\\scriptstyle {x^{2}}&\scriptstyle {+}&\scriptstyle {2y^{2}}&\scriptstyle {+}&\scriptstyle {6z^{2},}\\\scriptstyle {x^{2}}&\scriptstyle {+}&\scriptstyle {2y^{2}}&\scriptstyle {+}&\scriptstyle {8z^{2}.}\end{array}}$


The numbers which are not of these forms are the numbers


$\scriptstyle {4^{\lambda }(8\mu +7)}$ or $\scriptstyle {(8\mu +3)}$,
$\scriptstyle {4^{\lambda }(8\mu +3)}$,
$\scriptstyle {9^{\lambda }(9\mu +3)}$,
$\scriptstyle {4^{\lambda }(16\mu +14)}$, $\scriptstyle {(16\mu +6)}$, or $\scriptstyle {(4\mu +3)}$,
$\scriptstyle {4^{\lambda }(8\mu +5)}$,
$\scriptstyle {4^{\lambda }(8\mu +7)}$ or $\scriptstyle {(8\mu +5)}$.