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Is it true that for $$ f(x) = x^e\:mod\:m$$

$f$ is bijective and not the identify function if and only if

  1. $m$ is the product of unique primes
  2. For each prime factor of $m$ called $p$, $e$ is coprime to $p-1$
  3. Given $l=lcm(\{p_i-1\})$ (the smallest number which is a multiple of each factor of m) $e \neq 1\:mod\:l$

I've tried it for all combinations of e and m up to 500 and it appears to hold in those cases.

Irishmanluke
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  • See this question. – Bill Dubuque Jun 15 '21 at 01:02
  • I'm not quite following. I can see that showing that f is bijective is equivalent to showing that for all x<m: f(x) has unique eth roots mod m, however proving that part is not the part I'm having trouble with. I'm fairly sure that my criteria makes f bijective but I'm having trouble proving that it's necessary. It seems like the statement they're trying to prove would be really useful if it worked in the other direction. – Irishmanluke Jun 15 '21 at 16:11

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