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October 20th, 2015, 11:46 AM   #1
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Twin primes conjecture

Hi,

r=int(sqrt(n))

Conjecture : For each n>2 there is always at least one pair of twin-prime numbers between n!+n (not included) and n!+(r+1)^n (included)?

Any counterexample?
Is there a way to prove it if the conjecture is true?
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October 20th, 2015, 11:58 AM   #2
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It isn't even known if there are infinitely many twin primes, so you would probably need to prove that before you were able to tackle this. That has been unproved for quite a long time, so I assume the problem is very challenging.
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October 20th, 2015, 01:14 PM   #3
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I assume that there are infinite twin primes.
Trying to find a thin range where there at least one prime we can consider that as n goes to infinite (r+1)^n is far less < n!+n.
So m=(n!+(r+1)^n)/(n!+n) will be very < 2.
m will surely converges to some value between 1 and 2.
My goal as I said before if to find a thin range where there is at least one prime.
I do not think that my conjecture will work for a twin prime but for a prime I will not discard it.
Thank you for your comment.
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October 20th, 2015, 09:57 PM   #4
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Originally Posted by mobel View Post
Hi,

r=int(sqrt(n))

Conjecture : For each n>2 there is always at least one pair of twin-prime numbers between n!+n (not included) and n!+(r+1)^n (included)?
This is highly likely to be true, but beyond current mathematical technology to prove.
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October 20th, 2015, 10:00 PM   #5
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Quote:
Originally Posted by mobel View Post
I assume that there are infinite twin primes.
You can do that, and it's not unreasonable, but it doesn't help proving the conjecture at hand because (a priori) we don't know anything about how they might be distributed.

Quote:
Originally Posted by mobel View Post
Trying to find a thin range where there at least one prime we can consider that as n goes to infinite (r+1)^n is far less < n!+n.
So m=(n!+(r+1)^n)/(n!+n) will be very < 2.
m will surely converges to some value between 1 and 2.
Yes, it converges to 1.
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October 21st, 2015, 12:13 PM   #6
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Hard to prove but not impossible.
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October 21st, 2015, 01:13 PM   #7
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Hard to prove but not impossible.
Neither part of that statement has been proven -- but I tend to agree.
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