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Does anyone know how to do this problem, because i have no idea on this one
 
Does anyone know how to do this problem, because i have no idea on this one
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All I have so far is the base case. If you set n = 1 then you have a set with 2 (or n+1 = 1+1) positive integers where both integers have to be less than or equal to 2 (or 2*n = 2*1)  so the only option is that the set contains the numbers 1 and 2. For this set it is true that at least one integer in the set divides another integer in the set since 2 is divisible by 1.
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Does this sound right to anyone else?
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I'm not sure how to complete the inductive step.
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-Rachel
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I believe that is correct for the base case. I am also stuck on the inductive step. If we assume that k is true, then k+1 integers, each less than 2k, must have k or fewer prime numbers. I'm stuck on the next part. How do I prove that k+2 integers, each less than 2k+2, have fewer than k+1 prime numbers?
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--[[User:Jberlako|Jberlako]] 19:47, 21 January 2009 (UTC)
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http://batty.mullikin.org/uga_courses/math2610/spring03/induction.pdf  This website gives a very detailed explanation on how to solve it, I just noticed it tonight, I hope everyone gets to see it in time.  However I will note that at the bottom, the "Pigeon-hole Principle" I first thought this would be the easiest way to prove it, however it does not use induction.
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-Chris Ruderschmidt

Latest revision as of 18:24, 21 January 2009

Does anyone know how to do this problem, because i have no idea on this one


All I have so far is the base case. If you set n = 1 then you have a set with 2 (or n+1 = 1+1) positive integers where both integers have to be less than or equal to 2 (or 2*n = 2*1) so the only option is that the set contains the numbers 1 and 2. For this set it is true that at least one integer in the set divides another integer in the set since 2 is divisible by 1. Does this sound right to anyone else? I'm not sure how to complete the inductive step.

-Rachel

I believe that is correct for the base case. I am also stuck on the inductive step. If we assume that k is true, then k+1 integers, each less than 2k, must have k or fewer prime numbers. I'm stuck on the next part. How do I prove that k+2 integers, each less than 2k+2, have fewer than k+1 prime numbers?

--Jberlako 19:47, 21 January 2009 (UTC)


http://batty.mullikin.org/uga_courses/math2610/spring03/induction.pdf This website gives a very detailed explanation on how to solve it, I just noticed it tonight, I hope everyone gets to see it in time. However I will note that at the bottom, the "Pigeon-hole Principle" I first thought this would be the easiest way to prove it, however it does not use induction.

-Chris Ruderschmidt

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva