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4th power is -1 mod p

Let $$p$$ be a prime such that there exists an integer $$x$$ satisfying $x^4\equiv -1\pmod{p}$

Prove that $$p\equiv 1\pmod{8}$$

Source: Classic

Note by Daniel Liu
2 years, 3 months ago

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If $$x$$ satisfies $$x^4\equiv -1 \pmod p$$, then we claim that $$8$$ is the order of $$x$$, this will imply $$8|p-1$$.

Suppose the order is not $$8$$, then by properties of order it must be a divisor of $$8$$. This is absurd since either $$x\equiv -1$$ or $$x^2\equiv -1$$ would contradict $$x^4\equiv -1 \pmod p$$.

This can clearly be generalized for any powers of 2.

- 2 years, 3 months ago