What is the Proof for (p-1)!≡±1 (mod p) in Number Theory?

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SUMMARY

The discussion focuses on proving the congruence relation (p-1)!≡±1 (mod p) for prime numbers p≥3, as established by Wilson's Theorem. Participants outlined two key proofs: the first shows that if x is a solution to x²≡1 (mod p), then x must be congruent to ±1 (mod p). The second proof directly applies Wilson's Theorem, emphasizing the factorization of x²-1 into (x-1)(x+1) to demonstrate the relationship between factorials and modular arithmetic.

PREREQUISITES
  • Understanding of modular arithmetic
  • Familiarity with factorial notation and properties
  • Knowledge of Wilson's Theorem
  • Basic algebraic factorization techniques
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  • Study Wilson's Theorem in detail and its implications in number theory
  • Explore advanced topics in modular arithmetic, such as Chinese Remainder Theorem
  • Learn about the applications of factorials in combinatorial problems
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Mathematicians, number theorists, and students interested in advanced topics in modular arithmetic and factorial properties.

koukou
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Recall the definition of n! (read n factorial"):
n! = (n)(n-1)(n-2) ….(2)(1) =∏(k)
In both (a) and (b) below, suppose p≥3 is prime.
(a) Prove that if x∈ Zpx is a solution to x square ≡1 (mod p), then x ≡±1 (mod p).
(b) Prove that (p-1)!≡±1 (mod p)

Zpx x shoud be above p

a and b looks like some theorem proof
 
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Hints:
1. factorise x^2-1=(x-1)(x+1).

2. this theorem is called wilson theorem.
 
Hint: (2) can be looked at as a case of a and its inverse. The first part, (1) plays a special role in that.
 

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