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I have a task here with the expression $(m|p^n-n)$ and some quantifiers in front, where I need to pick the one that is true. The universal set is ${R}$. Edit: I read wrong, universal set is $(2,3,4...)$.

The four options are

  1. $∀m∀n∃p(m|p^n-n)$
  2. $∃m∃n∀p(m|p^n-n)$
  3. $∀m∃n∀p(m|p^n-n)$
  4. $∃m∀n∃p(m|p^n-n)$

I tried to look at $(m|p^n-n)$ as $p^n≡n(modm)$. In the first one, if $n=0$ and random $m=3$, we end up with $1≡0(mod3)$ and I think/hope that is enough to disprove that one. But is that the way to go? Just finding exceptions? I'm not quite sure how I need to think when solving a problem like this. Is thinking modulo correct? According to the answers 4 is the correct one, but I don't know how to prove/disprove these types of questions in general.

(Sorry for bad title, don't know how to formulate this question into nice title and English isn't my first language so all my "math language" is in another language)

confusedaf
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    I'm not sure if this helps, but my instinct is to rewrite each formula as plainly as possible; e.g., the first option as $∀m,n,∃p,k;\big(p^n=n+km\big).$ – ryang Dec 09 '21 at 18:43

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