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December 26th, 2012, 07:51 PM   #1
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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.
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December 26th, 2012, 08:13 PM   #2
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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.

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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
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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;






[color=beige]. . [/color]




[color=beige]. . [/color]


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

We cannot have an infinite sequence of consecutive composite numbers.


[color=beige]. . . [/color]

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