Strings of Composite Numbers

[Ender]

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.

[Rule]

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.

[Ender]

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.

[Rule]

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.