Question

Abstract Algebra:

The practice exam problem I have is "Prove that there are consecutive integers in a string of any desired (finite) length, all of which are not prime."

Answer 1

I should note that this is in the same section as Euclids Algorithm so I think the proof may involve use a form of that, but I really have no idea how to go about doing this.

Answer 2

My best idea so far is that this will involve a proof by contradiction, but, that I think will require a formula for generating an interval of length k between two primes, which seems like it will be impossible to just make up. Any thoughts from you clever math folks?

Answer 3

I have thought of a way that involves \(n!\)

Answer 4

as a short example

look at 5!

5!+2=[5*4*3+1]*2
5!+3=[5*4*2+1]*3
5!+4=[5*3*2+1]*4
5!+5=[4*3*2+1]*5

here is 4 consecutive numbers that are not prime

Answer 5

Hmmm, I think I follow. Is the idea then that you can chose any k for k! and add up to k?

Answer 6

k!+2
k!+3
...
k!+k

at least (k-1) consecutive non-primes

Answer 7

Ok thats what I thought you mean. I was testing numbers and saw 5!+7 is 127, a prime, so I wanted to make sure that we defined a stopping point. I am amazed you just came up with that. I was reading the wikipedia on the prime gap trying to figure this out. Is the idea then a proof by induction for k+1? If so I can probably handle that.

Answer 8

I don't think induction is needed....just choose any k...it clearly gives you at least k-1 consecutive nonprimes

Answer 9

Hah, quite right. Thanks very much for taking a look!