Sunday, April 22, 2007

Long stretches between primes

Theorem: For every positive integer n, there is a sequence of n consecutive positive integers containing no primes.

I felt I needed a hint, so I read a bit more of the proof:

Suppose n is a positive integer. Let x = (n + 1)! +2
Show that none of the numbers
x
x+1
x+2
x+3
......
x + (n-1)
is prime.

I wrote a test case:
n = 5
x = (5+1)!+2 = 722

so the 5 numbers that aren't prime
1. 722
2. 723 - is 3 * 241 (not-prime)
3. 724
4. 725
5.726

So, why on earth would this work?
Why is it true? Is it true for all cases?
How can I prove it?
This is going to be interesting!

Philosophical side note:
It is pretty amazing to sit and think about no matter how big n is that this theorem, once proved, will show that there is a consecutive list of numbers as big as n that are not prime. This makes me think of the universe and how at some point we have got to deal with the contradiction: Can the universe go on forever? Or can it end? If it ends, what is on the other side? I feel like neither one of these answers can be correct (1. it ends 2. it never ends) and that there is some fundamental problem with the way we see the universe.

Well, I'll leave the big question on the universe to God and I'll just worry about proving that there exists a long stretch of n consecutive numbers in our integer number system, which luckily can go on forever without any contradictions :)

cheers,
alex

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