Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/767

TABLE III.

Formule des nombres proposés $\ldots t^{2}+au^{2}.$

Formule de leurs diviseurs impairs, et premiers à $a\ldots py^{2}\pm 2qyz+rz^{2}$$=4an+b.$

${\begin{array}{|c|c|}\hline \\\scriptstyle {\mathrm {VALEURS\ DE} }&\scriptstyle {\mathrm {VALEURS\ CORRESPONDANTES\ DE} }\\&\overbrace {\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ \quad \qquad \qquad \qquad \qquad } \end{array}}$

${\begin{array}{|l|}&{\begin{array}{c|c|l}\ \ \quad a\ \quad &\quad p\ \quad &\qquad \qquad \qquad \qquad \qquad \qquad b\quad \qquad \qquad \qquad \qquad \qquad \\\hline \\1&\quad 1\quad &1\\2&1&1,3\\3&1&1,-5\end{array}}\\&{\begin{array}{cl|l}\quad 5\ \ \ &\left\{{\begin{array}{c|l}\ \ 1\quad \,\ &1,9\qquad \qquad \qquad \qquad \qquad \\\ \ 2\quad \,\ &3,7\end{array}}\right.\\\end{array}}\\&{\begin{array}{cl|l}\quad 6\ \ \ &\left\{{\begin{array}{c|l}\ \ 1\quad \,\ &1,7\\\ \ 2\quad \,\ &5,11\end{array}}\right.\\\end{array}}\\&{\begin{array}{c|c|l}\ \quad 7\ \ \quad &\quad 1\ \quad &1,9,11,-3,-5,-13\end{array}}\\&{\begin{array}{cl|l}\quad 10\ &\left\{{\begin{array}{c|l}\ \ 1\quad \,\ &1,9,11,19\\\ \ 2\quad \,\ &7,13,-3,-17\end{array}}\right.\\\end{array}}\\&{\begin{array}{c|c|l}\quad 11\ \quad &\ \,1,3\quad &1,3,5,9,15,-7,-13,-17,-19,-21\end{array}}\\&{\begin{array}{cl|l}\quad 13\ \,&\left\{{\begin{array}{c|l}\ \ 1\quad \,\ &1,9,17,25,-3,-93\\\ \ 2\quad \,\ &7,11,15,19,-5,-21\end{array}}\right.\\\end{array}}\\&{\begin{array}{cl|l}\quad 14\ \,&\left\{{\begin{array}{c|l}\ 1,2\ \ \ &1,9,15,23,25,-17\\3&3,5,13,19,27,-11\end{array}}\right.\\\end{array}}\\&{\begin{array}{cl|l}\quad 15\ \,&\left\{{\begin{array}{c|l}\ \ 1\quad \,\ &1,19,-11,-29\\\ \ 3\quad \,\ &17,23,-7,-13\end{array}}\right.\\\end{array}}\\&{\begin{array}{cl|l}\quad 17\ \,&\left\{{\begin{array}{c|l}\ 1,2\ \ \ &1,9,13,21,25,33,-15,-19\\3&3,7,11,28,27,31,-5,-29\end{array}}\right.\\\end{array}}\\&{\begin{array}{c|ll}\quad 19\ \quad &\ \ \ 1,4&\left\{{\begin{array}{ll}&1,5,7,11,17,28,25,85,-3,-13,-15,-21,\\&-27,-29,-31,-33,-37\end{array}}\right.\\\end{array}}\\&{\begin{array}{cl|l}\quad 21\ \,&\left\{{\begin{array}{c|l}\ \ 1\quad \,\ &1,25,37\\\ \ 2\quad \,\ &11,23,-13\\\ \ 3\quad \,\ &19,31,-29\\\ \ 5\quad \,\ &5,17,41\end{array}}\right.\\\end{array}}\\&{\begin{array}{cl|l}\quad 22\ \,&\left\{{\begin{array}{c|l}\ \ 1\quad \,\ &1,9,15,23,25,31,-7,-17,-39,-41\\\ \ 2\quad \,\ &13,19,21,29,35,43,-3,-5,27,-37\end{array}}\right.\\\end{array}}\\&{\begin{array}{c|ll}\quad 23\ \quad &\ \ \ 1,3&\left\{{\begin{array}{l}&1,3,9,13,25,27,29,31,35,39,41,-5,-7,-11\\&-15,-17,-19,-21,-33,-37,-43,-45\end{array}}\right.\\\end{array}}\\&{\begin{array}{cl|l}\quad 26\ \,&\left\{{\begin{array}{c|l}\ 1,3\ \ \ &1,3,9,17,25,27,35,43,49,51,-23,-29\\2,5&5,7,15,21,32,87,45,47,-11,-19,-33,-41\end{array}}\right.\\\end{array}}\\&{\begin{array}{cl|l}\quad 29\ \,&\left\{{\begin{array}{c|l}\ 1,5\ \ \ &1,5,9,13,95,33,45,49,53,57,-7,-23,-35,-51\\2,3&3,11,15,19,27,31,39,43,47,55,-17,-21,-37,-41\end{array}}\right.\\\end{array}}\\&{\begin{array}{cl|l}\quad 30\ \,&\left\{{\begin{array}{c|l}\ \ 1\,\ \quad &1,31,49,-41\\\ \ 2\quad \,\ &17,23,47,-7\\\ \ 3\quad \,\ &13,37,43,-53\\\ \ 5\quad \,\ &11,29,59,-19\end{array}}\right.\\\end{array}}\\\hline \end{array}}$