# Proving step to full proof

• October 1st 2010, 08:57 PM
I-Think
Proving step to full proof
I was doing a proof to prove that there exists arbitrarily long strings of composite numbers.
In more technical language, for every $m\in {N}$, there exists a n such that
$n, n+1, n+2, n+3,..., n+m$ is composite

to do so, I let $n=m!$,(for m>3) then we get the sequence:
$m!, m!+1, m!+2, m!+3,..., m!+m$

Every element of this sequence is obviously divisible by some integer, except $m!+1.$

Now I must prove that m!+1 is always an composite. I have no idea how to do this. I tried induction and contradiction.
• October 1st 2010, 11:53 PM
CaptainBlack
Quote:

Originally Posted by I-Think
I was doing a proof to prove that there exists arbitrarily long strings of composite numbers.
In more technical language, for every $m\element{N}$, there exists a n such that
$n, n+1, n+2, n+3,..., n+m$ is prime

to do so, I let $n=m!$,(for m>3) then we get the sequence:
$m!, m!+1, m!+2, m!+3,..., m!+m$

Every element of this sequence is obviously divisible by some integer, except $m!+1.$

Now I must prove that m!+1 is always an integer. I have no idea how to do this. I tried induction and contradiction.

There are so many typos in this I think you should correct them before expecting a reply.

Also this is number theory.

To show that for any $n\in \mathbb{N}$ there is a set of $n$ consecutive composites choose $m$ so that $m=n+1$.

Then $m!+2, m!+3,.. m!+m$ are all composite, and there are $m-1=n$ of these.

CB
• October 1st 2010, 11:59 PM
CaptainBlack
Quote:

Originally Posted by I-Think

Now I must prove that m!+1 is always an composite. I have no idea how to do this. I tried induction and contradiction.

You can't since if $m=2$ then $m!+1=5$ is prime

CB
• October 2nd 2010, 08:50 AM
I-Think
I apologize for the typos. I also apologize for posting this in the wrong forum but I did not expect the problem to involve Number Theory, as it is a topic I had not yet covered.

And as for proving that $m!+1$is always composite, I had stated it was over the range $m>3$
• October 2nd 2010, 10:40 AM
Traveller
Let n = (m+2)! + 2

P.S: m! + 1 is composite when m = p-1 for some prime p > 3.
• October 2nd 2010, 03:35 PM
CaptainBlack
Quote:

Originally Posted by i-think
i apologize for the typos. I also apologize for posting this in the wrong forum but i did not expect the problem to involve number theory, as it is a topic i had not yet covered.

And as for proving that $m!+1$is always composite, i had stated it was over the range $m>3$

ok 11!+1

cb
• October 4th 2010, 12:12 AM
aman_cc
Infact following should work

n = (m+1)! + 2