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 (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