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Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/767
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TABLE III.
Formule des nombres proposés
…
t
2
+
a
u
2
.
{\displaystyle \ldots t^{2}+au^{2}.}
Formule de leurs diviseurs impairs, et premiers à
a
…
p
y
2
±
2
q
y
z
+
r
z
2
{\displaystyle a\ldots py^{2}\pm 2qyz+rz^{2}}
=
4
a
n
+
b
.
{\displaystyle =4an+b.}
V
A
L
E
U
R
S
D
E
V
A
L
E
U
R
S
C
O
R
R
E
S
P
O
N
D
A
N
T
E
S
D
E
⏞
{\displaystyle {\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}}}
a
p
b
1
1
1
2
1
1
,
3
3
1
1
,
−
5
5
{
1
1
,
9
2
3
,
7
6
{
1
1
,
7
2
5
,
11
7
1
1
,
9
,
11
,
−
3
,
−
5
,
−
13
10
{
1
1
,
9
,
11
,
19
2
7
,
13
,
−
3
,
−
17
11
1
,
3
1
,
3
,
5
,
9
,
15
,
−
7
,
−
13
,
−
17
,
−
19
,
−
21
13
{
1
1
,
9
,
17
,
25
,
−
3
,
−
93
2
7
,
11
,
15
,
19
,
−
5
,
−
21
14
{
1
,
2
1
,
9
,
15
,
23
,
25
,
−
17
3
3
,
5
,
13
,
19
,
27
,
−
11
15
{
1
1
,
19
,
−
11
,
−
29
3
17
,
23
,
−
7
,
−
13
17
{
1
,
2
1
,
9
,
13
,
21
,
25
,
33
,
−
15
,
−
19
3
3
,
7
,
11
,
28
,
27
,
31
,
−
5
,
−
29
19
1
,
4
{
1
,
5
,
7
,
11
,
17
,
28
,
25
,
85
,
−
3
,
−
13
,
−
15
,
−
21
,
−
27
,
−
29
,
−
31
,
−
33
,
−
37
21
{
1
1
,
25
,
37
2
11
,
23
,
−
13
3
19
,
31
,
−
29
5
5
,
17
,
41
22
{
1
1
,
9
,
15
,
23
,
25
,
31
,
−
7
,
−
17
,
−
39
,
−
41
2
13
,
19
,
21
,
29
,
35
,
43
,
−
3
,
−
5
,
27
,
−
37
23
1
,
3
{
1
,
3
,
9
,
13
,
25
,
27
,
29
,
31
,
35
,
39
,
41
,
−
5
,
−
7
,
−
11
−
15
,
−
17
,
−
19
,
−
21
,
−
33
,
−
37
,
−
43
,
−
45
26
{
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
29
{
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
30
{
1
1
,
31
,
49
,
−
41
2
17
,
23
,
47
,
−
7
3
13
,
37
,
43
,
−
53
5
11
,
29
,
59
,
−
19
{\displaystyle {\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}}}