Exercise 2.6.6 (Solve $x^3 + x^2 - 4 \equiv 0 \pmod{7^3}$)

Question

Solve $x^3 + x^2 - 4 \equiv 0 \pmod{7^3}$.

Richard Ganaye · Accepted Answer

\begin{proof} 
The values of $f(a)$ for $0 \leq a <7$ are congruent modulo $7$ to
$$-4, -2, 1, 4, 6, 6, 3.$$
Therefore the congruence $x^3 + x^2 - 4 \equiv 0 \pmod{7}$ has no solution. A fortiori $x^3 + x^2 - 4 \equiv 0 \pmod{7^3}$ has no solution.
\end{proof}

Exercise 2.6.6 (Solve $x^3 + x^2 - 4 \equiv 0 \pmod{7^3}$)

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Exercise 2.6.6 (Solve $x^3 + x^2 - 4 \equiv 0 \pmod{7^3}$)

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