Search   Memberlist   Usergroups
 Page 1 of 1 [2 Posts]
Author Message
Proginoskes
science forum Guru

Joined: 29 Apr 2005
Posts: 2593

Posted: Fri Jul 14, 2006 4:54 am    Post subject: Re: Prine Number thought experiment

 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
science forum beginner

Joined: 22 Nov 2005
Posts: 14

Posted: Fri Jul 14, 2006 3:31 am    Post subject: Prine Number thought experiment

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!

 Display posts from previous: All Posts1 Day7 Days2 Weeks1 Month3 Months6 Months1 Year Oldest FirstNewest First
 Page 1 of 1 [2 Posts]
 The time now is Sun Nov 18, 2018 12:10 am | All times are GMT
 Jump to: Select a forum-------------------Forum index|___Science and Technology    |___Math    |   |___Research    |   |___num-analysis    |   |___Symbolic    |   |___Combinatorics    |   |___Probability    |   |   |___Prediction    |   |       |   |___Undergraduate    |   |___Recreational    |       |___Physics    |   |___Research    |   |___New Theories    |   |___Acoustics    |   |___Electromagnetics    |   |___Strings    |   |___Particle    |   |___Fusion    |   |___Relativity    |       |___Chem    |   |___Analytical    |   |___Electrochem    |   |   |___Battery    |   |       |   |___Coatings    |       |___Engineering        |___Control        |___Mechanics        |___Chemical

 Topic Author Forum Replies Last Post Similar Topics how to deduce a number validation algorithm? Tim_Mac Math 1 Fri Jul 21, 2006 10:43 am a subset of natural number levine121323@yahoo.com Math 1 Fri Jul 21, 2006 6:17 am Another look at triangle number factoring. Dan11 Math 2 Tue Jul 18, 2006 7:05 pm Hofstadter's _GEB_ -- Help with number theory Daniel al-Autistiqui Math 3 Tue Jul 18, 2006 5:33 pm My thought on F=ma Euler Cheung Physics 9 Sun Jul 16, 2006 2:44 am