Natural Blogarithms

I was browsing through some unread blogs in my RSS feeder and saw something that I thought was interesting, so I thought I’d pass it along.

From think again!

- I read somewhere that there are an infinite number of them, but at the same time I have found areas where there are none!
- What do you mean?
- I have found one hundred consecutive integers none of which is a prime!
- What do you mean by consecutive?
- They follow one after the other. For example 32, 33, 34, 35, 36 are five consecutive integers none of which are prime.
- I see. But you have found one hundred consecutive integers and none of them is a prime?
- Did I say one hundred? I meant one hundred thousand!
- You are pulling my leg right?
- I thought I was pulling the left.

It turns out that the longest stretch of non-prime, or composite, integers in a row that has been found so far is 1,442. This is based on a search of numbers up to 10^18. (http://hjem.get2net.dk/jka/math/primegaps/maximal.htm). But someone posted an even better answer in the comments of the think again! website. It’s fairly easy to show that for any natural number, n, there exists a prime gap of that size, that is, n consecutive composite integers:

Theorem: For any natural number, n, there exists a set of n consecutive integers such that none are prime.

Proof: Let n be a natural number. Clearly (n+1)! + 2 is divisible by 2, since both (n+1)! and 2 are divisible by 2. By the same reasoning:
(n+1)! + 3 is divisible by 3
(n+1)! + 4 is divisible by 4
…
(n+1)! + n is divisible by n
(n+1)! + (n+1) is divisible by n+1

Thus the numbers from (n+1)! +2 up to (n+1)! + (n+1) represent a list of n consecutive integers, none of which are prime. Q.E.D.

I thought that was pretty slick.

Trackback URI | Comments RSS