Author Topic: Strings of Composite Numbers  (Read 4235 times)

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Offline Ender

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Strings of Composite Numbers
« on: May 23, 2007, 11:06:09 am »
Prove that there exists a string of n consecutive composite numbers for any number n. In other words, prove that there are strings of numbers as long as you like that haven't a single prime in them.

Proof will be posted next Wednesday unless requested otherwise.


Offline Rule

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Re: Strings of Composite Numbers
« Reply #1 on: May 23, 2007, 01:44:39 pm »
Consider for example, n!+9, n!+10, n!+11, n!+12, ..., n! + n
where n is any natural number >= 9.

We have a sequence of n-9 consecutive non-prime integers.  Since n-9 is unbounded, the desired result follows.

Offline Ender

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Re: Strings of Composite Numbers
« Reply #2 on: May 23, 2007, 08:02:10 pm »
Yep =) that's right.

The factorial is the key move. In the form n! + a, if you can factor a out and make the sum strictly a product of numbers, then n! + a obviously has to be composite. Oh, and I probably should have clarified in my post of the problem that "composite" implies we're doing this over the natural numbers.
« Last Edit: May 23, 2007, 08:06:59 pm by Ender »

Offline Rule

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Re: Strings of Composite Numbers
« Reply #3 on: May 23, 2007, 08:16:18 pm »
Oh, and I probably should have clarified in my post of the problem that "composite" implies we're doing this over the natural numbers.

I didn't know what a composite number was, but
In other words, prove that there are strings of numbers as long as you like that haven't a single prime in them.
made the question quite clear. 

I like how the solution isn't very fussy.  :)