D`UN THÉORÈME NOUVEAU

n
1.2.3.4.5. . . (n−1) + 1
n
1,2,3, . . . n −1
n n −1
n−1
n= 3,1.2 + 1 = 3,
n= 5,1.2.3.4 + 1 = 25,
n= 7,1.2.3.4.5.6 + 1 = 721 = 7.103,
n= 11,1.2.3. . . 10 + 1 = 3628801 = 11.329891,
n= 13,1.2.3. . . 12 + 1 = 479001601 = 13.36846277,
, .
(x+ 1)(x+ 2)(x+ 3)(x+ 4) . . . (x+n−1),
x
(x+ 1)(x+ 2)(x+ 3)(x+ 4) . . . (x+n−1)
=xn−1+A0xn−2+A00 xn−3+A000 xn−4+...+A(n−1),
A0, A00 , A000 , . . .
x+ 1 x
(x+ 2)(x+ 3)(x+ 4)(x+ 5) . . . (x+n)
= (x+ 1)n−1+A0(x+ 1)n−2+A00 (x+ 1)n−3+A000 (x+ 1)n−4+...+A(n−1) ;
x+ 1 x+n
(x+n)(xn−1+A0xn−2+A00 xn−3+A000 xn−4+...+A(n−1))
= (x+ 1)n+A0(x+ 1)n−1+A00 (x+ 1)n−2+A000 (x+ 1)n−3+. . . +A(n−1)(x+ 1),

x
xn+ (n+A0)xn−1+ (nA0+A00 )xn−2+ (nA00 +A000 )xn−3+. . .
=xn+ (n+A0)xn−1+·n(n−1)
2+ (n−1)A0+A00 ¸xn−2
+·n(n−1)(n−2)
2.3+(n−1)(n−2) 2A0+ (n−2)A00 +A000 ¸xn−2+. . . .
n+A0=n+A0,
nA0+A00 =n(n−1)
2+ (n−1)A0+A00 ,
nA00 +A000 =n(n−1)(n−2)
2.3+(n−1)(n−2) 2A0+ (n−2)A00 +A000 ,
............................................................... ,
A0=n(n−1)
2,
2A00 =n(n−1)(n−2)
2.3+(n−1)(n−2) 2A0,
3A000 =n(n−1)(n−2)(n−3)
2.3.4+(n−1)(n−2)(n−3) 2.3A0+(n−2)(n−3) 2A00 ,
A0, A00 , A000 , . . .
1,2,3, . . . n −1
A(n−1) 1.2.3.4. . . (n−1)
A0, A00 , A000 , . . .
n
A0, A00 , A000 , . . . A(n−2) n A(n−1)
n
n(n−1)
2,n(n−1)(n−2) 2.3, . . . , (n−1)(n−2) 2,(n−1)(n−2)(n−3) 2.3, . . .
n
n n −1n
n(n−1)
1.2,n(n−1)(n−2) 1.2.3, . . .
n
n(n−1)(n−2) . . . 1
1.2.3. . . n
n
n
n
◦A0n2A00 3A000 ,4A , . . . (n−2)A(n−2)
A0, A00 , A000 , . . . , A(n−2)
n n

◦A(n−1) n
(n−1)A(n−1) =n(n−1)(n−2) . . . 1
1.2.3. . . n
+(n−1)(n−2) . . . 1
1.2. . . (n−1) A0+(n−2)(n−3) ...1 1.2. . . (n−2) A00 +. . . ,
(n−1)A(n−1) = 1 + A0+A00 +A000 +. . . +A(n−2) ;
An−1+ 1 = nA(n−1) −A0−A00 −A000 −. . . −A(n−2) ;
A0, A00 , . . . , A(n−2) n A(n−1) + 1 n
1.2.3.4. . . (n−1) + 1
n n
◦x
(x+ 1)(x+ 2)(x+ 3) . . . (x+n−1) −xn−1+ 1
n n
◦xn−1n x n
(x+ 1)(x+ 2)(x+ 3) . . . (x+n−1) + 1
n x = 0
◦x n x =µn +ρ ρ
n
x+ 1, x + 2, x + 3, . . . , x +n−1
n
(x+ 1)(x+ 2)(x+ 3) ...(x+n−1)
n−xn−1+ 1 xn−1−1
n
n−1, n −2, n −3, . . .
n
−1,−2,−3, . . . ,
n−1, n −2, . . .
1.2.3. . . (n−1);
1.2.3. . . (n−1) + 1,
1.2.3. . . (n−2) −1,
1.22.3...(n−3) + 1,
1.22.32.4...(n−4) −1,
. . . . . . . . . . . . . . . . . . . . ,

n
·1.2.3. . . µn−1
2¶¸2
±1
nn−1
2
n−1
2
◦n−1
2= 2m n = 4m+ 1 (1.2.3. . . 2m)2+ 1 n
4m+ 1
n n 4m+ 1
◦n−1
2= 2m−1n= 4m−1 [1.2.3. . . (2m−1)]2−1n
[1.2.3. . . (2m−1)]2−1 = [1.2.3. . . (2m−1) + 1][1.2.3. . . (2m−1) −1];
n
1.2.3...(2m−1) + 1 1.2.3. . . (2m−1) −1,
n
1.2.3. . . (2m−1) ±1
n
n
n n n
2,3, . . . , n −1n
1.2.3. . . (n−1) + 1
n2,3,4, . . . , n −1
1.2.3. . . (n−1)
1.2.3. . . (n−1) + 1
n
2,3,4, . . . , n −2n1
n4m+ 1 4m−1n
2,3,4. . . , 2m n −1n−1
2,3,4,..., 2m−1n1−1
n
n
1,2,3, . . . , n (n−1)
(n−1)
nn−1−(n−1)(n−1)n−1+(n−1)(n−2)
2(n−2)n−1
−(n−1)(n−2)(n−3)
2.3(n−3)n−1+. . . + 1;

1,2n−1,3n−1, . . .
(n−1)
1,2,3. . . , n −1
1.2.3.4. . . (n−1) =nn−1−(n−1)(n−1)n−1+(n−1)(n−2)
2(n−2)n−1
−(n−1)(n−2)(n−3)
2.3(n−3)n−1+. . . + 1.
n
nn−10
(n−1)n−1,(n−2)n−2, . . .
0,1,1, . . .
−(n−1) + (n−1)(n−2)
2−(n−1)(n−2)(n−3) 2.3+. . . ,
(1 −1)n−1−1,−1;
1.2.3. . . (n−1) n−1
1.2.3. . . (n−1) + 1
n n
◦6
3
6m, 6m±1,6m±2,6m±3;
6m6m±2 6m±1,6m±3
3
3 3 6m±1
p−a, p, p +a
3
6m±1
a6n6n±2
a= 6n±2p= 6m+ 1
p+a= 6(m+n) + 3 = 6(m+n)−1
p−a= 6(m−n)−1 = 6(m−n) + 3;
p+a p −a6µ±1p= 6m−1
p+a= 6(m+n) + 1 = 6(m+n)−3
p−a= 6(m−n)−3 = 6(m−n) + 1;
p+a p −a6µ±1a
6n±2a6n6