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

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Discussion Overview

The discussion revolves around a mathematical expression involving modular arithmetic and prime numbers, specifically the equation [((n-1)/2)]^2 = -(-1)^[(n-1)/2] (mod n). Participants explore the implications of Wilson's Theorem in relation to this expression and seek clarification on the problem posed.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested, Homework-related

Main Points Raised

  • Some participants assert that p is prime and not equal to 2, while others question the presence of p in the original expression.
  • One participant suggests that the expression should be interpreted as (mod p).
  • Another participant states that p equals n, prompting further inquiry into the question posed.
  • A participant emphasizes the importance of understanding Wilson's Theorem in solving the problem and provides historical context about Wilson's contributions.
  • One participant expresses a desire to prove the equation for every prime n, excluding 2, using Wilson's Theorem but indicates they have not made significant progress.
  • A later reply indicates that the original poster has resolved their question independently.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of the expression or the application of Wilson's Theorem. Multiple competing views remain regarding the problem's setup and the necessary conditions for its proof.

Contextual Notes

Some assumptions regarding the definitions of variables and the context of the theorem remain unresolved, and there are indications of missing mathematical steps in the discussion.

xax
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p is prime, not 2. Thanks in advance
 
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xax said:
p is prime, not 2. Thanks in advance
There is no p in your expression!
 
he had to have meant (mod p)
 
p = n
 
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!
 
xax said:
p = n


:smile:

So what is your question and what have you tried to solve it?
 
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).
 
Nevermind guys, I figured it out. Thanks for your input.
 

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