Difference between revisions of "Fermat's Little Theorem"
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| − | A frequently used corollary of Fermat's | + | A frequently used corollary of Fermat's Little Theorem is <math> a^p \equiv a \pmod {p}</math>. |
As you can see, it is derived by multipling both sides of the theorem by a. The restated form is nice because we no longer need to restrict ourselves to integers <math>{a}</math> not divisible by <math>{p}</math>. | As you can see, it is derived by multipling both sides of the theorem by a. The restated form is nice because we no longer need to restrict ourselves to integers <math>{a}</math> not divisible by <math>{p}</math>. | ||
Revision as of 09:04, 11 July 2006
Fermat's Little Theorem is highly useful in number theory for simplifying computations in modular arithmetic (which students should study more at the introductory level if they have a hard time following the rest of this article).
Statement
If 
is an integer, 
is a prime number and is not divisible by , then 
.
Note: This theorem is a special case of Euler's Totient Theorem.
Corollary
A frequently used corollary of Fermat's Little Theorem is 
.
As you can see, it is derived by multipling both sides of the theorem by a. The restated form is nice because we no longer need to restrict ourselves to integers not divisible by .
Sample Problem
One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that 
. Find the value of 
. (AIME 1989/9)
By Fermat's Little Theorem, we know 
is congruent to 
modulo 5. Hence,
Continuing, we examine the equation modulo 3,
Thus, is divisible by three and leaves a remainder of four when divided by 5. It's obvious that 
, so the only possibilities are 
or 
. It quickly becomes apparent that 174 is much too large, so must be 144.
Credit
This theorem is credited to Pierre de Fermat.
