Find the solution prime number p is divisir of cubic equation

Asked Jun 17 '22 at 15:38

Active Jun 17 '22 at 21:56

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Find all prime numbers p for which there exists a unique a ∈ {1, 2, . . . , p} such that $a^3 − 3a + 1$ is divisible by p.

When I see the solution of this problem , the first think I surprised was we need to find the root of this equation, and I think this isn't related to some normal solution .

My question is how root of this cubic equation is related to the solution divisible by p ? enter image description here

edited Jun 17 '22 at 21:56

asked Jun 17 '22 at 15:38

Le Nam

Try it first with, say $p=2$ and $p=3$. – Dietrich Burde Jun 17 '22 at 15:54
$p\mid f(a)\iff x=a,$ is a root of $f(x)$ in $\Bbb Z_p,,$ that's the "root" view of the problem. Is that clear? – Bill Dubuque Jun 17 '22 at 15:59
What is "the solution of this problem" that you see? – Robert Israel Jun 17 '22 at 16:09
@RobertIsrael I have added it, but I don't understand much the solution – Le Nam Jun 17 '22 at 22:01

Find the solution prime number p is divisir of cubic equation

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