[((n-1)/2)]^2 = -(-1)^[(n-1)/2] (mod n)

  • Thread starter xax
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In summary, the conversation is about proving that p is prime, not 2, using Wilson's Theorem. The person asking the question also mentions that Wilson was not a mathematician but credited for the theorem. Afterwards, they provide an attempt at solving the problem using the theorem. Eventually, the person figures out the solution on their own.
  • #1
xax
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p is prime, not 2. Thanks in advance
 
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  • #2
xax said:
p is prime, not 2. Thanks in advance
There is no p in your expression!
 
  • #3
he had to have meant (mod p)
 
  • #4
p = n
 
  • #5
This is not a difficult problem, but you must understand Wilson's Theorem. Wilson, by the way, was never a mathematician, and as a student went on to become a lawyer.

He is credited with noticing the theorem, but was unable to prove it. So much for the immortal glory of a name theorem!
 
  • #6
xax said:
p = n


:rofl:

So what is your question and what have you tried to solve it?
 
  • #7
I thought it was clear what the question was(since the others understood): prove that this is true for every n prime, n!= 2. Using the Wilson theorem, I didn't go far:
((n-1)/2)!*((n+1)/2)*((n+3)/2)*...*(n-1) = -1 (mod n).
 
  • #8
Nevermind guys, I figured it out. Thanks for your input.
 

1. What is the meaning of the equation [((n-1)/2)]^2 = -(-1)^[(n-1)/2] (mod n)?

This equation is known as the Wilson's theorem and it states that for any prime number n, the product of all numbers from 1 to n-1 is congruent to -1 modulo n.

2. How is this equation used in mathematics?

Wilson's theorem is primarily used in number theory and has applications in cryptography, primality testing, and the study of perfect numbers. It is also used to prove other theorems in number theory.

3. What are the conditions for this equation to hold true?

For this equation to hold true, n must be a prime number. If n is composite, the equation will not be satisfied.

4. Can this equation be generalized for non-prime numbers?

No, Wilson's theorem only holds true for prime numbers. For non-prime numbers, there are other similar theorems such as the Wilson's quotient theorem, but they have different conditions and equations.

5. How is this equation connected to other mathematical concepts?

Wilson's theorem is connected to various mathematical concepts such as modular arithmetic, group theory, and field theory. It is also closely related to other theorems in number theory such as Fermat's little theorem and Euler's theorem.

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