Page:Joseph Louis de Lagrange - Œuvres, Tome 2.djvu/474

et l’on aura à résoudre l’équation

$6=r^{2}+46s^{2}.$

Or, puisque $\mathrm {B} =46,\ \mathrm {E} =6,\ e=2,$ on aura (43) $\lambda <{\frac {{\sqrt {46}}+2}{6}}$ et $>{\frac {{\sqrt {46}}+2}{6}}-1\,;$ donc

$l=1\quad {\text{et}}\quad \varepsilon =\lambda \mathrm {E} -e=4.$

Ayant $\varepsilon ,$ on formera (31 et 33) les séries suivantes, où le signe $<$ indique qu’il faut prendre les nombres entiers qui sont immédiatement plus petits,

${\begin{alignedat}{3}\mathrm {E} \ \ \ &=6,&&&\varepsilon \quad &=\ \ 4,\\\mathrm {E} _{1}\ \,&={\frac {46-16}{6}}=\ \ 5,&\lambda _{1}\ \,&<{\frac {{\sqrt {46}}+4}{5}}=\ \,2,&\varepsilon _{1}\ \,&=\ \ 2.\ \ 5-4=6,\\\mathrm {E} _{2}\ \,&={\frac {46-36}{5}}=\ \ 2,&\lambda _{2}\ \,&<{\frac {{\sqrt {46}}+6}{2}}=\ \ 6,&\varepsilon _{2}\ \,&=\ \ 6.\ \ 2-6=6,\\\mathrm {E} _{3}\ \,&={\frac {46-36}{2}}=\ \ 5,&\lambda _{3}\ \,&<{\frac {{\sqrt {46}}+6}{5}}=\ \,2,&\varepsilon _{3}\ \,&=\ \ 2.\ \ 5-6=4r,\\\mathrm {E} _{4}\ \,&={\frac {46-16}{5}}=\ \ 6,&\lambda _{4}\ \,&<{\frac {{\sqrt {46}}+4}{5}}=\ \,1,&\varepsilon _{4}\ \,&=\ \ 1.\ \ 6-4=2,\\\mathrm {E} _{5}\ \,&={\frac {46-\ \ 4}{6}}=\ \ 7,&\lambda _{5}\ \,&<{\frac {{\sqrt {46}}+2}{5}}=\ \,1,&\varepsilon _{5}\ \,&=\ \ 1.\ \ 7-2=5,\\\mathrm {E} _{6}\ \,&={\frac {46-25}{7}}=\ \ 3,&\lambda _{6}\ \,&<{\frac {{\sqrt {46}}+5}{5}}=\ \,3,&\varepsilon _{6}\ \,&=\ \ 3.\ \ 3-5=4,\\\mathrm {E} _{7}\ \,&={\frac {46-16}{3}}=10,&\lambda _{7}\ \,&<{\frac {{\sqrt {46}}+4}{5}}=\ \,1,&\varepsilon _{7}\ \,&=\ \ 1.10-4=6,\\\mathrm {E} _{8}\ \,&={\frac {46-36}{10}}=\ \ 1,&\lambda _{8}\ \,&<{\frac {{\sqrt {46}}+6}{5}}=12,&\varepsilon _{8}\ \,&=12.\ \ 1-6=6,\\\mathrm {E} _{9}\ \,\ \,&={\frac {46-36}{1}}=10,&\lambda _{9}\ \,&<{\frac {{\sqrt {46}}+6}{5}}=\ \,1,&\varepsilon _{9}\ \,&=\ \ 1.10-6=4,\\\mathrm {E} _{10}&={\frac {46-16}{10}}=\ \ 3,&\lambda _{10}&<{\frac {{\sqrt {46}}+4}{5}}=\ \,3,&\varepsilon _{10}&=\ \ 3.\ \ 3-4=5,\\\mathrm {E} _{11}&={\frac {46-25}{3}}=\ \ 7,&\lambda _{11}&<{\frac {{\sqrt {46}}+5}{5}}=\ \,1,&\varepsilon _{11}&=\ \ 1.\ \ 7-5=2,\\\mathrm {E} _{12}&={\frac {46-\ \ 4}{7}}=\ \ 6,&\lambda _{12}&<{\frac {{\sqrt {46}}+2}{5}}=\ \,1,&\varepsilon _{12}&=\ \ 1.\ \ 6-2=4,\\\mathrm {E} _{13}&={\frac {46-16}{6}}=\ \ 5,\quad &\lambda _{13}&<{\frac {{\sqrt {46}}+4}{5}}=\ \,2,\quad &\varepsilon _{13}&=\ \ 2.\ \ 5-4=6,\\\ldots &\ldots \ldots \ldots \ldots \ldots ,&\ldots &\ldots \ldots \ldots \ldots \ldots ,&\ldots &\ldots \ldots \ldots \ldots \ldots .\end{alignedat}}$