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:

Fermat's Little Theorem

Fermat's little theorem gives a condition that a prime must satisfy:

Theorem. If P is a prime, then for any integer A,

( AP - A ) must be divisible by P.

Let's check:
29 - 2 = 510, is not divisible by 9, so it cannot be prime.
35 - 3 = 240, is divisible by 5, because 5 is prime.

Presentation Suggestions:
This may be a good time to explain the difference between a necessary and sufficient condition.

The Math Behind the Fact:
This theorem can be used as a way to test if a number is not prime, although it cannot tell you if a number is prime.

Fermat's theorem is a special case of a result known as Euler's theorem: that for any positive integer N, and any integer A relatively prime to N:

( Aphi(N) - A ) must be divisible by N,

where phi(N) is Euler's totient function that returns the number of positive integers less than or equal to N that are relatively prime to N. So when N is prime, phi(N)=N.

Fermat's "little" theorem should not be confused with Fermat's Last Theorem.

How to Cite this Page:
Su, Francis E., et al. "Fermat's Little Theorem." Math Fun Facts. <>.

Keywords:    number theory, primality test
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!