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I understand the proof and I can use Fermat's little theorem, but I'm having trouble getting an intuition of why this theorem is true. The version I know is this one (I know there are some variations of the theorem that are a bit more general but I'm less interested as they are probably even harder to fully grasp).

If $p$ is a prime and a is any integer not divisible by $p$, then $a^{p − 1} − 1$ is divisible by $p$.

Would anyone have a little drawing in mind or a little explanation that would make me click? I feel it tremendously helps to have an intuition of "why it works". Thanks!

Gary
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FluidMechanics Potential Flows
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  • You might easily get it from a more general theorem, which is Euler's Theorem. See p. 15 of this material for a proof. – soupless Mar 09 '22 at 11:12
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    @soupless Do you think that Euler's theorem can be understood intuitively better than Fermat's little theorem ? – Peter Mar 09 '22 at 11:15
  • This duplicate gives some proofs, which are more conceptual and therefore give more intuition - it uses Lagrange (I agree with the comment below). – Dietrich Burde Mar 09 '22 at 11:16
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    In my opinion, the best way to get intuition about Fermat's little theorem is to learn a little bit of group theory and Lagrange's theorem. – TheSilverDoe Mar 09 '22 at 11:16
  • I guess I might need to learn a bit of group theory and more fundamental maths because I'm very far from understanding Euler's Theorem and its proof or the duplicate linked. I was sort of looking for an intuition "with the hands" accessible to maths undergraduates or students from high school but there might not be one? – FluidMechanics Potential Flows Mar 09 '22 at 11:23
  • Every math undergraduate learns Lagrange's theorem early on in a course in modern/abstract algebra, but I agree that you can't just jump into wikipedia's explanation if you don't know any group theory. Just for confirmation, can you share which proof of Fermat you already know and weren't satisfied by? – Mark S. Mar 09 '22 at 12:28
  • It is in French but I will translate it in English and include it in the post when I have some time later today. – FluidMechanics Potential Flows Mar 09 '22 at 13:30
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    Did you follow the link in the second answer of that other thread? It gives several proofs for the theorem, many of which are quite simple, particularly the combinatorics proof. – Paul Sinclair Mar 10 '22 at 03:35
  • I didn't, and I'm going to look at it, and maybe I'll find another proof more intuitive than the one I currently have. Thanks for pointing it out ;) – FluidMechanics Potential Flows Mar 10 '22 at 14:09

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