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© 1999-2010 by Francis Edward Su
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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. <http://www.math.hmc.edu/funfacts>.

Keywords:    number theory, primality test
Subjects:    number theory
Level:    Easy
Fun Fact suggested by:   Lesley Ward
Suggestions? Use this form.
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