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# proof of wilson's theorem by differentiation on fermat's little theorem? watch

1. Does it work? I'm trying to do it, but keep getting to (p-1)! (three horizontal lines) 0 (mod p), which obviously isn't right.

Could you show how? Bearing in mind I'm an A level student with very little further maths knowledge
2. You simply cannot differentiate like that with modulo arithmetic. Take for example:

is a solution but differentiating the equation as you seem to have done gives

for which the solution doesn't work. Moduluar arithmetic is that little bit more tricky, you need to turn a modular equivalence into an equality for differnetiating
3. Basically, the set your function operates on, , doesn't have enough structure to support differentiation. The proof of Wilson's theorem is actually quite easy, and boils down to the observation that every number other than has a unique multiplicative inverse which is not the number itself.
4. I read somewhere that Wilson's Theorem "follows directly from Fermat's Little Theorem".
If differentiation like that isn't a correct approach using the Little Theorem as starting point, what is?
5. I'm not saying there isn't a link, but I'm not really seeing it. The normal approach is as Zhen describes (and in fact, Wilson's theorem is in many ways "simpler" than Fermat's Little Theorem).
6. (Original post by DFranklin)
I'm not saying there isn't a link, but I'm not really seeing it. The normal approach is as Zhen describes (and in fact, Wilson's theorem is in many ways "simpler" than Fermat's Little Theorem).
The first proof I learnt was more complicated, but does indeed start with Fermat's Little Theorem.

Spoiler:
Show
We know that for all , so by various properties of which I cannot recall, . Now consider .
7. (Original post by Zhen Lin)
The first proof I learnt was more complicated, but does indeed start with Fermat's Little Theorem.

Spoiler:
Show
We know that for all , so by various properties of which I cannot recall, . Now consider .
Oh yes, thanks. Though you could argue it starts with observations about roots of polynomials in .

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