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Prine Number thought experiment
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Proginoskes
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Joined: 29 Apr 2005
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PostPosted: Fri Jul 14, 2006 4:54 am    Post subject: Re: Prine Number thought experiment Reply with quote

robert.w.adams@verizon.net wrote:
Quote:
Suppose I am given N buckets, and in each bucket I have a collection of
prime numbers from 2 to Pk inclusive, where Pk is the kth prime. Each
bucket can have a different number of primes (a different k), but must
include all primes from 2 to Pk without skipping any. Now suppose that
I form a number by taking one prime from each bucket and multiplying
these primes to form a number S.

My goal is to insure that all numbers that can be formed from this
process are greater than some integer L and less than some integer M,
without any "holes" between L and M.

I assume that means "every number between L and M is obtainable".

Quote:
This must be done by specifying the largest prime in each bucket, as
well as the number of buckets. Intuition would suggest that the 1st
bucket has the largest # of primes, and each bucket thereafter would
have a decreasing number.

But the buckets are all symmetric; if you swap the first bucket with
the last bucket, you have the same set of numbers.

Quote:
My guess is that there is no solution to this problem. Any thoughts?

It's impossible unless you consider one of the trivial solutions (one
bucket with 2 and 3, or more than one bucket, all containing only 2).
This is because of Chebyshev's Theorem, which states that for all
integers N, there is a prime number between N and
2 * N.

PROOF: If you have more than one bucket, the products will all be
composite. If one bucket contains 2 and 5, and you pick a prime p_i
from each of the other buckets, then two possible values of S are

S(1) = 2 * p_2 * p_3 * ... * p_N, and
S(2) = 5 * p_2 * p_3 * ... * p_N.

However, S(2) >= 2 * S(1), so there is a prime number P between S(1)
and S(2), which S can never equal.

If you have two buckets with 2 and 3 in them, and choose a prime p_i
from each of the other buckets, you can do something similar, except
now
S(1) = 2 * 2 * p_3 * ... * p_N, and
S(2) = 3 * 3 * p_3 * ... * p_N,

and S(2) >= 2 * S(1) here as well.

So, if you have more than one bucket, all but one of them contains only
a 2. If your last bucket has a 2 and a 3 in it, you run into problems.
(I'll let you work this one out.)

If you only have one bucket, then you clearly can't have a 5 in it.
Again, I'll leave the details for you.

--- Christopher Heckman
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robert.w.adams@verizon.ne
science forum beginner


Joined: 22 Nov 2005
Posts: 14

PostPosted: Fri Jul 14, 2006 3:31 am    Post subject: Prine Number thought experiment Reply with quote

Suppose I am given N buckets, and in each bucket I have a collection of
prime numbers from 2 to Pk inclusive, where Pk is the kth prime. Each
bucket can have a different number of primes (a different k), but must
include all primes from 2 to Pk without skipping any. Now suppose that
I form a number by taking one prime from each bucket and multiplying
these primes to form a number S.

My goal is to insure that all numbers that can be formed from this
process are greater than some integer L and less than some integer M,
without any "holes" between L and M.
This must be done by specifying the largest prime in each bucket, as
well as the number of buckets. Intuition would suggest that the 1st
bucket has the largest # of primes, and each bucket thereafter would
have a decreasing number.

My guess is that there is no solution to this problem. Any thoughts?

Thanks!

Bob Adams
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