My Math Forum Infinitely long consecutive composite numbers.

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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
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Re: Infinitely long consecutive composite numbers.

Quote:
 Originally Posted by agentredlum You should replace the word 'infinitely' with another word like 'arbitrarily' for example, otherwise your request contradicts the infinitude of primes.
Yeah very true ^^;

infinitely -> arbitrarily

December 26th, 2012, 09:03 PM   #4
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Re: Infinitely long consecutive composite numbers.

Hello, suugakuedu!

The problem is not clearly stated.

Quote:
 Show that there exist infinitely long consecutive composite numbers.

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;

$\text{Consider }n!\text{ and the next }n\text{ integers.}$

$\text{W\!e have: }\:n!\,+\,k\;\text{ for }k=0\text{ to }n.$

$\text{While }n!\,+\,1\text{ may be prime, we know that the rest:}$
[color=beige]. . [/color]$n!\,+\,2,\; n!\,+\,3,\;\cdots,\;n!\,+\,n\;\text{ are all composite.}$

$\text{Hence, we have a sequence of }n-1\text{ consecutive composite integers.}$

$\text{As }n\text{ gets larger, we have longer and longer sequences}$
[color=beige]. . [/color]$\text{of consecutive composite integers.}$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

We cannot have an infinite sequence of consecutive composite numbers.

$\text{This would involve the absurdity }\infty!\,\text{ and its absurd followers:}$
[color=beige]. . . [/color]$\infty!\,+\,2,\;\infty!\,+\,3,\;\infty!\,+\,4,\;\c dots\;\infty!\,+\,\infty$

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