# Page:On the expression of a number in the form ππ₯Β²+ππ¦Β²+ππ§Β²+ππ’Β².djvu/2

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12
Mr Ramanujan, On the expression of a number
1. If ${\displaystyle \scriptstyle {d>14}}$, ${\displaystyle \scriptstyle {14}}$ is an exception; and so
 ${\displaystyle \scriptstyle {2\leq d\leq 14}}$ (2Β·421).

(2Β·43)β${\displaystyle \scriptstyle {x^{2}+y^{2}+3z^{2}+du^{2}}}$.

If ${\displaystyle \scriptstyle {d>6}}$, ${\displaystyle \scriptstyle {6}}$ is an exception; and so
 ${\displaystyle \scriptstyle {3\leq d\leq 6}}$ (2Β·431).

(2Β·44)β${\displaystyle \scriptstyle {x^{2}+2y^{2}+2z^{2}+du^{2}}}$.

If ${\displaystyle \scriptstyle {d>7}}$, ${\displaystyle \scriptstyle {7}}$ is an exception; and so
 ${\displaystyle \scriptstyle {2\leq d\leq 7}}$ (2Β·441).

(2Β·45)β${\displaystyle \scriptstyle {x^{2}+2y^{2}+3z^{2}+du^{2}}}$.

If ${\displaystyle \scriptstyle {d>10}}$, ${\displaystyle \scriptstyle {10}}$ is an exception; and so
 ${\displaystyle \scriptstyle {3\leq d\leq 10}}$ (2Β·451).

(2Β·46)β${\displaystyle \scriptstyle {x^{2}+2y^{2}+4z^{2}+du^{2}}}$.

If ${\displaystyle \scriptstyle {d>14}}$, ${\displaystyle \scriptstyle {14}}$ is an exception; and so
 ${\displaystyle \scriptstyle {4\leq d\leq 14}}$ (2Β·461).

(2Β·47)β${\displaystyle \scriptstyle {x^{2}+2y^{2}+5z^{2}+du^{2}}}$.

If ${\displaystyle \scriptstyle {d>10}}$, ${\displaystyle \scriptstyle {10}}$ is an exception; and so
 ${\displaystyle \scriptstyle {5\leq d\leq 10}}$ (2Β·471).

We have thus eliminated all possible sets of values of ${\displaystyle \scriptstyle {a}}$, ${\displaystyle \scriptstyle {b}}$, ${\displaystyle \scriptstyle {c}}$, ${\displaystyle \scriptstyle {d}}$, 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