x²+x+1=0
Δ=1-4=-3
√Δ=√-3=i√3
[tex]x_1=\frac{-1+i\sqrt{3} }{2} \\\\x_2=\frac{-1-i\sqrt{3} }{2} \\\\\\\varepsilon=\frac{-1+i\sqrt{3} }{2}[/tex]
[tex]\varepsilon^2=\frac{-2-2i\sqrt{3} }{4} =\frac{-2(1+i\sqrt{3}) }{4} =-\frac{1+i\sqrt{3} }{2}[/tex]
[tex]\varepsilon^3=\frac{1+3}{4} =1[/tex]
[tex]\varepsilon^3=element\ neutru \implies \varepsilon^4=\varepsilon\\\\\varepsilon^5=\varepsilon^2\\\\\varepsilon^6=\varepsilo1\\\\...\\\varepsilon^2^0^1^3=1[/tex]
2013:3=671 rest 0
S=1+ε+ε²+1+ε+ε²+...+1+ε+ε²+1
Avem 2014 termeni
2014:3=671 rest 1
Deci avem 671 grupe de 1+ε+ε² plus restul 1, adica termenul 1
S=671(1+ε+ε²)+1
Calculam 1+ε+ε²
[tex]1+\varepsilon+\varepsilon^2=1+\frac{-1+i\sqrt{3} }{2} -\frac{1+i\sqrt{3} }{2} =1+\frac{-2}{2} =1-1=0[/tex]
S=671×0+1=1
