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 February 8th, 2011, 04:15 PM #1 Senior Member   Joined: Jan 2011 Posts: 560 Thanks: 1 Particular isolated primes Usually they define isolated primes by some value (more or less than 6 for example). Trying to solve a problem I have found some isolated numbers using a formula instead of values. I called those numbers les 'nombres premiers insulaires'. A prime number P is called insulaire if there are no prime numbers in its vicinity (+ or - 2*int(ln(p)). Example p=211 2*int(ln(211))=10 All the numbers from 211+k(k=1 to 10) are composite All the numbers from 211-k (k=1 to 10) are composite So the sequence of those numbers is U(p)= (211,2179,2513,3967, and so on) I will come back to those numbers because I feel something interesting is hiding there.
 February 8th, 2011, 04:57 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Particular isolated primes I've calculated the first million such primes. Would you like my list?
 February 8th, 2011, 05:08 PM #3 Senior Member   Joined: Jan 2011 Posts: 560 Thanks: 1 Re: Particular isolated primes Just the first 1000. It will be okay. Thank you.
 February 8th, 2011, 05:10 PM #4 Senior Member   Joined: Jan 2011 Posts: 560 Thanks: 1 Re: Particular isolated primes I want to know if there is some bounded k such as p+k*int(ln(p) and p-k*int(ln(p) (both prime-free).
February 8th, 2011, 05:27 PM   #5
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Re: Particular isolated primes

Quote:
 Originally Posted by Bogauss Just the first 1000.
Code:
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February 8th, 2011, 05:30 PM   #6
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Re: Particular isolated primes

Quote:
 Originally Posted by Bogauss I want to know if there is some bounded k such as p+k*int(ln(p) and p-k*int(ln(p) (both prime-free).
For every k > 0 there are infinitely many p such that $[p-k\lfloor\ln p\rfloor,\, p+k\lfloor\ln p\rfloor]$ contains exactly one prime.

February 8th, 2011, 05:37 PM   #7
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Re: Particular isolated primes

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by Bogauss I want to know if there is some bounded k such as p+k*int(ln(p) and p-k*int(ln(p) (both prime-free).
For every k > 0 there are infinitely many primes p such that $[p-k\lfloor\ln p\rfloor,\, p+k\lfloor\ln p\rfloor]$ contains exactly one prime.
And if k=p!

February 8th, 2011, 05:40 PM   #8
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Re: Particular isolated primes

Quote:
Originally Posted by Bogauss
Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by Bogauss I want to know if there is some bounded k such as p+k*int(ln(p) and p-k*int(ln(p) (both prime-free).
For every k > 0 there are infinitely many primes p such that $[p-k\lfloor\ln p\rfloor,\, p+k\lfloor\ln p\rfloor]$ contains exactly one prime.
And if k=p!
The result doesn't hold, since it assumes that k is constant.

You could show that there are infinitely many primes q such that [q - p! log q, q + p! log q] contains only one prime, but not that [p - p! log p, p + p! log p] contains just one prime. (This latter result is, of course, false for p >= 2.)

 February 8th, 2011, 06:04 PM #9 Senior Member   Joined: Jan 2011 Posts: 560 Thanks: 1 Re: Particular isolated primes Let k=1 then there is always at least one prime between : p and p+2*ln(p) assuming that ln(p) is almost equal to ln(p -ln(p))
February 8th, 2011, 06:14 PM   #10
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Re: Particular isolated primes

Quote:
 Originally Posted by Bogauss Let k=1 then there is always at least one prime between : p and p+2*ln(p) assuming that ln(p) is almost equal to ln(p -ln(p))
This is false. First, it's not even remotely similar to what I said -- I'm talking about an interval around a prime containing zero other primes and you're talking about forcing an interval to contain at least one prime. Second, it fails for p = 7, 113, 139, 199, 211, 293, 317, 523, 773, 839, 863, and 8580 other primes below a million.

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