Gaps in Primes

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 to see why. Suppose I want to find (N-1) 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 (N-1) 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. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Lesley Ward

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