Competition problem — Number theory

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Number theory Difficulty 6.0 AIME, harder Prove it

13. Let $p$ be a prime, $2 \nmid \delta_{p}(a)$ . Prove: The congruence equation $a^{x}+1 \equiv 0(\bmod p)$ has no solution.

Solution

13. Let $p-1=2^{l} \cdot c, 2 \nmid c$ , by property IV of $\S 3$ and $2 \nmid \delta_{p}(a)$ , we know that $2^{l}$ must divide the index of $a$ (with any primitive root $g$ as the base). From this and the fact that the index of -1 is $(p-1) / 2$ , the desired conclusion can be derived.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.