Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
No subject limitations
Search only in selected subjects
    Calculus or Analysis
    Number Theory
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

From the Fun Fact files, here is a Fun Fact at the Easy level:

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 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. <>.

Subjects:    number theory
Level:    Easy
Fun Fact suggested by:   Lesley Ward
Suggestions? Use this form.
Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!