My Math Forum Conjecture about consecutive primes

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 September 16th, 2015, 05:11 AM #1 Banned Camp   Joined: Dec 2013 Posts: 1,117 Thanks: 41 Conjecture about consecutive primes Hi eveybody, For any finite sequence of consecutive primes P(n) (n>=2) of length l > 2 it always exists at least one even positive integer 2m such as : 2m = P(n)+C(j) with C(j) is composite for any P(n) and m < p(1)*p(2)*....*p(n) Example : m=49 2m=98 setP(n)=(3,5,7,11,13} m<3*5*7*11*13 3+95=98 5+93=98 7+91=98 11+87=98 13+85=98 All the C(j)={95,93,91,87,85} are composite. Any counterexample? Any proof? Thanks for your comments.
 September 16th, 2015, 05:46 AM #2 Math Team   Joined: Apr 2010 Posts: 2,778 Thanks: 361 For a proof you could try to show a prime gap of p(n) - p(1) exists below p(1) * ... * p(n) for all n > 1. Thanks from mobel
 September 16th, 2015, 05:47 AM #3 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 Let me see if I understand the problem. You have a sequence of 3 or more consecutive (odd?) primes and you want to know if for any such there is a positive integer m less than the product of the primes such that any of the primes plus 2m is composite. Right?
September 16th, 2015, 05:53 AM   #4
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Quote:
 Originally Posted by CRGreathouse Let me see if I understand the problem. You have a sequence of 3 or more consecutive (odd?) primes and you want to know if for any such there is a positive integer m less than the product of the primes such that any of the primes plus 2m is composite. Right?
Right!
3 or more consecutive odd primes.

For sure odds P(2)=3 is the starting point
2 is prime for sure but it will be superfluous to indicate it because 2+something = 2m then something will always be composite.

 September 16th, 2015, 05:58 AM #5 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 If I understand correctly it seems highly likely. You're looking for a small number of composites with a large range to search. With 3 primes starting at p the 'chance' of a prime given a nearby odd is ~2/log p so the chance for three composites is around (1 - 2/log p)^3, which is more than 1/2 once you hit a few tens of thousands. For p above 10^6 you have > 10^18 chances, each more than 50% likely to work. As you go higher or increase the number of primes it only gets easier. Thanks from mobel
September 16th, 2015, 06:14 AM   #6
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Quote:
 Originally Posted by CRGreathouse If I understand correctly it seems highly likely. You're looking for a small number of composites with a large range to search. With 3 primes starting at p the 'chance' of a prime given a nearby odd is ~2/log p so the chance for three composites is around (1 - 2/log p)^3, which is more than 1/2 once you hit a few tens of thousands. For p above 10^6 you have > 10^18 chances, each more than 50% likely to work. As you go higher or increase the number of primes it only gets easier.
Yes but this is not a proof.
Even if something is highly likely does not mean that it always work except if you find a mathematical proof.

 September 16th, 2015, 09:49 AM #7 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 Sure -- just checking that we're all on the same page here. I checked that there are no counterexamples of triples less than 10^9, so there's no need to worry about small cases, just the general/large cases.
 September 17th, 2015, 02:20 AM #8 Senior Member   Joined: Dec 2007 Posts: 687 Thanks: 47 Let $p_1,\ldots,p_n$ be a list of consecutive odd primes, where $n>2$ and $\displaystyle P=\prod_{i=1}^np_i$. Clearly $P+1$ is even. Now, notice that $\frac{P+1}{2}+\frac{P-3}{2}=P-12$ prime numbers in sequence; (3) this satisfies the conditions of the problem, that for any product $P$ of at least three odd primes we find \$N
September 17th, 2015, 03:20 AM   #9
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Quote:
 Originally Posted by CRGreathouse ...any of the primes plus 2m is composite.
Probably doesn't change much about a proof, but that's not how I read the problem. I read it as '...any of the primes plus a composite is 2m.'

September 17th, 2015, 05:51 PM   #10
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Quote:
 Originally Posted by Hoempa Probably doesn't change much about a proof, but that's not how I read the problem. I read it as '...any of the primes plus a composite is 2m.'
Yes Hoempa, I read Mobel in the same way, CRG probably made a typo.

Anyway, this won't change the conclusion I wrote above after small modifications.

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