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December 26th, 2012, 07:51 PM  #1 
Newbie Joined: Dec 2012 From: Japan (pure Japanese) Posts: 11 Thanks: 0  Infinitely long consecutive composite numbers.
Show that there exist infinitely long consecutive composite numbers.  it's a good math trivia. Many of you might know this already. It's hard to come up with an idea, but the question itself isn't hard at all. 
December 26th, 2012, 08:13 PM  #2 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: Infinitely long consecutive composite numbers.
You should replace the word 'infinitely' with another word like 'arbitrarily' for example, otherwise your request contradicts the infinitude of primes. 
December 26th, 2012, 08:19 PM  #3  
Newbie Joined: Dec 2012 From: Japan (pure Japanese) Posts: 11 Thanks: 0  Re: Infinitely long consecutive composite numbers. Quote:
infinitely > arbitrarily  
December 26th, 2012, 09:03 PM  #4  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Re: Infinitely long consecutive composite numbers. Hello, suugakuedu! The problem is not clearly stated. Quote:
I don't think you are referring to "infinitely long composite numbers" . . . a silly concept. You may mean: There is an infinitely long sequence of consecutive composite numbers. [color=beige]. . [/color]But even that is vague and indeterminate. If my interpretation is correct, you might say instead: [color=beige]. . [/color]There is a sequence of consecutive composite numbers of any desirable length. We can construct such a sequence like this; [color=beige]. . [/color] [color=beige]. . [/color] ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ We cannot have an infinite sequence of consecutive composite numbers. [color=beige]. . . [/color]  

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