Number Theory | Wilson's Theorem and Classification of Primes

We give a Corollary to Wilson's Theorem as well as a classification of primes using this result.

[PunmasterSTP]

Another great video. These are so great that they should be shown on…prime-time!

[SanketAlekar]

Loved this method of proof, it was great and feels more elegant. What I did (I always try and prove on my own before watching all the way through) was not as elegant but I think works.

n prime => (n-2)! = 1 mod n by corollary

For the other way, I decided to prove the inverse. n not prime => (n-2)! is not 1 mod n.

n not prime => n = a*b for two numbers less than n which are not 1 or n (and also n-1). Which means that a and b are numbers between 1.... n-2 and thus a term a*b appears in (n-2)! so n divides (n-2)! so (n-2)! = 0 mod n which is not 1 mod n.

[georgesadler]

Professor Penn, thank you for another fine analysis of Wilson's Theorem and all the Classification of Primes.

[arnavgupta]

What a beauty very much fascinated by these proofs u bring thank u ! 🙏🙏

[maxmustermann]

What does it mean for a modolo to be negative? Sorry for the noob question, but I always thought the remainder of two positive numbers has to be positive?

[viswanathank]

I have a new formula.
Fact(p-1)is congruent to 0 mod(p**2 - 1)