Solve $5x^3 -2x + 1 \equiv 0 $ mod 243.
We're using Hensel Lifting theorem to solve. I am trying to use the example we did in class but I am not following. I know we need to the dericative and find the prime factorization of 243.
2026-02-23 08:22:03.1771834923
Elementary Number Theory: Cubic Congruences
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You’ll never solve this congruence modulo $3^5$, because there isn’t even a solution modulo $3$. For, over $\Bbb F_3$, your polynomial is $2x^3+x+1=-(x^3-x-1)$. Why? Because for $n\in\Bbb Z$, $n^3\equiv n\pmod3$, and as a function on $\Bbb F_3$, your polynomial is constant nonzero.