congruencies

teorema:
Si p€P ==> n^{p} =[p]= n
demostració:
(n+1)^{p}=n^{p}+pk+1=(npk+n)+pk+1=(n+1)pk+(n+1)


teorema:
Si p€P ==> n^{p+(-1)} =[p]= 1
demostració:
(n+1)^{p+(-1)}=(n^{p}+pk+1)/(n+1)=(n·n^{p+(-1)}+pk+1)/(n+1)=...
(n+1)^{p+(-1)}=(n(pk+1)+pk+1)/(n+1)=((n+1)pk+(n+1))/(n+1)=pk+1


proposició:
Si p€P ==> 1^{p+(-1)}+...+n^{p+(-1)} =[p]= n


proposició:
Si p€P ==> 1^{p}+...+n^{p} =[p]= ( n(n+1)/2 )


teorema:
ax =[a·m]= ab <==> x =[m]= b
demostració:
ax=a·mt+ab
x=mt+b




3x =[9]= 18 <==> x =[3]= 6 =[3]= 0
2x =[8]= 18 <==> x =[4]= 9 =[4]= 1
3x =[6]= 18 <==> x =[2]= 6 =[2]= 0
2x =[6]= 18 <==> x =[3]= 9 =[3]= 0


teorema:
Si ( p€P & p=2k+1 ) ==> ax^{2} =[p]= a <==> x =[p]= a^{(p+(-1))/2}
demostració:
(a^{(p+(-1)/2)})^{2} = a^{p+(-1)} =[p]= 1
a^{p} =[p]= a


2x^{2} =[5]= 2 <==> x =[5]= 4
3x^{2} =[5]= 3 <==> x =[5]= 9 =[5]= 4
4x^{2} =[5]= 4 <==> x =[5]= 16 =[5]= 1


teorema:
Si ( p€P & p=2k+1 ) ==> ax^{2} =[p]= (-a) <==> x =[p]= a^{(p+(-1))/2}·i
demostració:
(a^{(p+(-1)/2)}·i)^{2} = (-1)a^{p+(-1)} =[p]= (-1)
(-1)a^{p} =[p]= (-a)


2x^{2} =[5]= (-2) <==> x =[5]= 4i
3x^{2} =[5]= (-3) <==> x =[5]= 9i =[5]= 4i
4x^{2} =[5]= (-4) <==> x =[5]= 16i =[5]= i


teorema:
Si ( p€P & p=3k+1 ) ==> ax^{3} =[p]= a <==> x =[p]= a^{(p+(-1))/3}
demostració:
(a^{(p+(-1)/3)})^{3} = a^{p+(-1)} =[p]= 1
a^{p} =[p]= a


2x^{3} =[7]= 2 <==> x =[7]= 4
3x^{3} =[7]= 3 <==> x =[7]= 9 =[7]= 2
4x^{3} =[7]= 4 <==> x =[7]= 16 =[7]= 2


teorema:
Si p€N ==> ...
...x^{2}+2bx =[p^{2}]= (-1)b^{2} <==> x =[p]= (-b)
demostració:
x^{2}+2bx =[p^{2}]= (-1)b^{2}
x^{2}+2bx+b^{2} =[p^{2}]= 0
(x+b)^{2} =[p^{2}]= 0
x+b =[p]= 0
x =[p]= (-b)


x^{2}+4x =[9]= (-4) <==> x =[3]= (-2) =[3]= 1
x^{2}+6x =[9]= (-9) <==> x =[3]= (-3) =[3]= 0
x^{2}+8x =[9]= (-16) <==> x =[3]= (-4) =[3]= (-1) =[3]= 2