Difference between revisions of "Fermat's Little Theorem"
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Note: This theorem is a special case of [[Euler's Totient Theorem]]. | Note: This theorem is a special case of [[Euler's Totient Theorem]]. | ||
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+ | == Proof == | ||
+ | Let <math>S = \{1,2,3,\cdots, p-1\}</math>. Then, we note that <math>S = \{1a, 2a, \cdots, (p-1)a\} \pmod{p}</math>. Clearly none of the <math>ai</math> may be divisible by <math>p</math>, so it suffices to show that all of the elements in the latter set are distinct. If <math>ai \equiv aj \pmod{p}</math> for <math>\text{gcd}\, (a,p) = 1</math>, then by the cancellation rule, <math>i \equiv j \pmod{p}</math>. Thus, the product of the elements of <math>S</math> written in two ways, taken <math>\mod{p}</math>, gives <math>1a \cdot 2a \cdots (p-1)a \equiv 1 \cdot 2 \cdots p-1 \pmod{p}</math>. Cancelling again, we have <math>a^{p-1} \equiv 1 \pmod{p}</math>. | ||
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+ | A similar version can be used to prove [[Euler's Totient Theorem]], if we let <math>S = \{\text{natural numbers relatively prime and less than}\ n\}</math>. | ||
== Corollary == | == Corollary == |
Revision as of 14:23, 28 January 2009
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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.
Proof
Let . Then, we note that . Clearly none of the may be divisible by , so it suffices to show that all of the elements in the latter set are distinct. If for , then by the cancellation rule, . Thus, the product of the elements of written in two ways, taken , gives . Cancelling again, we have .
A similar version can be used to prove Euler's Totient Theorem, if we let .
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 Fermat.