Michael's Notes & Blog

Is $ 2018^{1991}+2018^{1990}+1 $ prime or not

\begin{equation} \begin{aligned} a^{1991} + a^{1990} + 1 \ &= a^{1991} + a^{1990} + 1 + a^{1989} - a^{1989} \ &= (a^2 + a + 1)a^{1989} - (a^{1989} - 1) \ &= (a^2 + a + 1)a^{1989} - (a^3 - 1)(a^{1989} - 1) / (a^3 - 1) \ &= (a^2 + a + 1)a^{1989} - (a - 1)(a^2 + a + 1)(a^{1989} - 1)/(a^3 - 1) \ &= (a^2 + a + 1)(a^{1989} - (a - 1)(a^{1989} - 1)/(a^3 - 1)) \ &= (a^2 + a + 1)(a^{1989} - (a - 1)({a^3}^{663} - 1)/(a^3 - 1)) \ &= (a^2 + a + 1)(a^{1989} - (a - 1)(A^{663} - 1)/(A - 1)) \ &= (a^2 + a + 1)(a^{1989} - (a - 1)(A^{662} + A^{661} + ... + 1)) \end{aligned} \end{equation}

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