Wilson theorem Question Explanation

In summary, Wilson's theorem is a mathematical theorem discovered by John Wilson in 1770. It states that a natural number is a prime number if and only if it satisfies the condition that if a natural number p is prime, then (p-1)! + 1 is divisible by p. This theorem is significant because it provides a simple and efficient way to determine if a number is prime or not. However, it cannot be used to find all prime numbers as it only provides a necessary condition, not a sufficient one.
  • #1
mathsss2
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0
How do I explain this:

Let [tex]p[/tex] be odd prime explain why: [tex]2*4*...*(p-1)\equiv (2-p)(4-p)*...*(p-1-p)\equiv(-1)^{\frac{(p-1)}{2}}*1*3*...*(p-2)[/tex] mod [tex]p[/tex].



Relevant equations

Gauss lemma
wilson's theorem [[tex](p-1)!\equiv-1[/tex] mod[tex] p[/tex]]



The attempt at a solution
pairing? need assistance


Thanks
 
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  • #2
Hint: What are [itex]2-(2-p)[/itex], [itex]4-(4-p)[/itex], ... [itex](p-1)-(p-1-p)[/itex]? Are they divisible by [itex]p[/itex]?
 
  • #3
Thanks, this problem is solved.
 

What is Wilson's theorem?

Wilson's theorem is a mathematical theorem that states that a natural number is a prime number if and only if it satisfies a specific condition.

What is the condition stated in Wilson's theorem?

The condition stated in Wilson's theorem is that if a natural number p is prime, then (p-1)! + 1 is divisible by p.

Who discovered Wilson's theorem?

Wilson's theorem was discovered by mathematician John Wilson in 1770.

What is the significance of Wilson's theorem?

Wilson's theorem is significant because it provides a simple and efficient way to determine if a number is prime or not.

Can Wilson's theorem be used to find all prime numbers?

No, Wilson's theorem cannot be used to find all prime numbers as it only provides a necessary condition for a number to be prime, not a sufficient one. It is possible for a number to satisfy the condition but still not be prime.

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