Proving Twin Primes with n and n+2 mod n(n+2)

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In summary, the conversation discusses proving the equation (n+1)! = 2(n-1)! mod n+2, as well as using this to prove that if 4[(n-1)! + 1] + n = 0 mod n(n+2) holds, then n, n+2 are twin primes. The converse of this theorem is also proven, making it a necessary and sufficient condition for (n, n+2) to be a pair of twin primes. The use of Wilson's theorem is also mentioned.
  • #1
SomeRandomGuy
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1.) Show that (n+1)! = 2(n-1)! mod n+2
I finished this one. Actually very easy.

2.) Let n > 2 be odd. Prove that if 4[(n-1)! + 1] + n = 0 mod n(n+2) holds, then n, n+2 are twin primes. Hint says to use the previous problem.

I don't even know what to do for this problem.

3.) Prove the converse of the theorem in the preceeding problem is also true. Thus, the two problems together constitute a necessary and sufficient condition for (n, n+2) to be a pair of twin primes.

Obviously, if I can figure out #2, this one will be a walk in the park.

Thanks for any help given, I appreciate it.
 
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  • #2
For 2), consider 4[(n-1)! + 1] + n = 0 modulo n and modulo n+2 separately. Remember Wilson's theorem.
 
  • #3
Alright, I think I got it. Let me just make sure before moving on, Here is what I did:

(1) 4(n-1)!+4+n = 0 mod n
(2) 4(n-1)!+4+n = 0 mod n+2

(1) tells us that (n-1)! = -1 mod n since n = 0 mod n. So, by Wilson's theorem, n is prime. There was a little more work with the second equation, but came to the result of (n+1)! = -1 mod n+2 and made a similar conclusion. Is this correct?
 
  • #4
Looks good. Part 3) should be no problem now?
 
  • #5
shmoe said:
Looks good. Part 3) should be no problem now?

I haven't looked at it, but our Professor said the first and 3rd questions are really easy. If I have any questions i'll post on here. Thanks for your help man, I really do appreciate it.
 

1. What are twin primes?

Twin primes are a pair of prime numbers that differ by 2, such as 41 and 43.

2. How does the "n and n+2 mod n(n+2)" method prove twin primes?

The "n and n+2 mod n(n+2)" method involves checking if n and n+2 have a common factor (other than 1) by performing the modulus operation on n(n+2). If they do not have a common factor, then they are both prime numbers and therefore twin primes.

3. What is the significance of using n and n+2 in this method?

The use of n and n+2 is important because it ensures that the numbers being tested are always 2 apart, which is the definition of twin primes.

4. Are there any limitations to this method?

Yes, this method can only be used to prove that a pair of numbers are twin primes. It cannot be used to generate an infinite list of twin primes.

5. Has this method been proven to be accurate?

Yes, this method has been mathematically proven to accurately identify twin primes. However, it is not the most efficient method for finding twin primes.

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