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

2. If $\scriptstyle {d>14}$, $\scriptstyle {14}$ is an exception; and so

   $\scriptstyle {2\leq d\leq 14}$

   (2·421).

   
   

   (2·43) $\scriptstyle {x^{2}+y^{2}+3z^{2}+du^{2}}$.

   If $\scriptstyle {d>6}$, $\scriptstyle {6}$ is an exception; and so

   $\scriptstyle {3\leq d\leq 6}$

   (2·431).

   
   

   (2·44) $\scriptstyle {x^{2}+2y^{2}+2z^{2}+du^{2}}$.

   If $\scriptstyle {d>7}$, $\scriptstyle {7}$ is an exception; and so

   $\scriptstyle {2\leq d\leq 7}$

   (2·441).

   
   

   (2·45) $\scriptstyle {x^{2}+2y^{2}+3z^{2}+du^{2}}$.

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

   $\scriptstyle {3\leq d\leq 10}$

   (2·451).

   
   

   (2·46) $\scriptstyle {x^{2}+2y^{2}+4z^{2}+du^{2}}$.

   If $\scriptstyle {d>14}$, $\scriptstyle {14}$ is an exception; and so

   $\scriptstyle {4\leq d\leq 14}$

   (2·461).

   
   

   (2·47) $\scriptstyle {x^{2}+2y^{2}+5z^{2}+du^{2}}$.

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

   $\scriptstyle {5\leq d\leq 10}$

   (2·471).

   
   

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

   1,	1,	1,	1	1,	2,	3,	5	1,	2,	4,	8
   1,	1,	1,	2	1,	2,	4,	5	1,	2,	5,	8
   1,	1,	2,	2	1,	2,	5,	5	1,	1,	2,	9
   1,	2,	2,	2	1,	1,	1,	6	1,	2,	3,	9
   1,	1,	1,	3	1,	1,	2,	6	1,	2,	4,	9
   1,	1,	2,	3	1,	2,	2,	6	1,	2,	5,	9
   1,	2,	2,	3	1,	1,	3,	6	1,	1,	2,	10
   1,	1,	3,	3	1,	2,	3,	6	1,	2,	3,	10
   1,	2,	3,	3	1,	2,	4,	6	1,	2,	4,	10
   1,	1,	1,	4	1,	2,	5,	6	1,	2,	5,	10
   1,	1,	2,	4	1,	1,	1,	7	1,	1,	2,	11
   1,	2,	2,	4	1,	1,	2,	7	1,	2,	4,	11
   1,	1,	3,	4	1,	2,	2,	7	1,	1,	2,	12
   1,	2,	3,	4	1,	2,	3,	7	1,	2,	4,	12
   1,	2,	4,	4	1,	2,	4,	7	1,	1,	2,	13
   1,	1,	1,	5	1,	2,	5,	7	1,	2,	4,	13
   1,	1,	2,	5	1,	1,	2,	8	1,	1,	2,	14
   1,	2,	2,	5	1,	2,	3,	8	1,	2,	4,	14
   1,	1,	3,	5