Vanished

The quartic equation $a_1 x^4 + a_2 x^3 + a_3 x^2 + a_4 x + a_5 = 0$ is obviously a biquadratic equation if $(a_2, a_4) = (0,0)$ .

However, if $(a_2, a_4) = (0,0)$ is not satisfied, then this same quartic equation can be converted into a biquadratic equation if

$m a_2 ^3 + n a_1 ^2 a_4 - p a_1 a_2 a_3 = 0$ is satisfied, where $m,n,p$ are coprime constant integers.

Calculate $m^6 + n^6 - p^6$ .

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This is part of the series: " It's easy, believe me! "