Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/768

TABLE IV.

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