We know there are infinitely many primes,
so are many interesting questions you can
ask about the distribution of primes,
i.e., how they spread themselves out.
Here is something to ponder:
are there arbitrarily large "gaps"
in the sequence of primes?
At first this may seem
like a tough question to tackle, since it is sometimes
tedious to determine whether a number is prime.
But it may help to look at the problem a different way:
can I find long sequences of successive integers
which are all composite?
Yes, and now it is easy to see why.
Suppose I want to find (N1) consecutive integers that
are composite. The number N! has, as factors, all
numbers between 1 and N. Therefore:
N!+2 is composite, since it is divisible by 2.
N!+3 is composite, since it is divisible by 3.
In fact, for similar reasons,
N!+k is composite for all k between 2 and N.
This is a string of (N1) successive integers which
are all composite.
Presentation Suggestions:
It may be good to warm up by asking is what the
largest prime gap less than 100.
The Math Behind the Fact:
Sometimes simple deductions can lead to surprising
results!
How to Cite this Page:
Su, Francis E., et al. "Gaps in Primes."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
