Given two primes $11$ and $5$, find all $α > 1$ such that $α^{5} \equiv 1 \mod 11$.
How would you compute that?
2026-03-28 04:22:23.1774671743
Equation solution in modular arithmetic
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A small table will do: $$\begin{array}{r*{5}{c}} x &\pm2&\pm3&\pm4&\pm5\\ \hline x^2& 4&-2&5&3\\ x^4 &5&4&3&-2\\ x^5&\mp1&\pm1&\pm1&\pm1\\ \hline \end{array}$$ Hence the solutions are $$\alpha= 1,\;-2,\;3,\;4,\;5. $$