Editing Fermat primality test (section)

==Concept==
[[Fermat's little theorem]] states that if ''p'' is prime and ''a'' is not divisible by ''p'', then

:<math>a^{p-1} \equiv 1 \pmod{p}.</math>

If one wants to test whether ''p'' is prime, then we can pick random integers ''a'' not divisible by ''p'' and see whether the congruence holds. If it does not hold for a value of ''a'', then ''p'' is composite. This congruence is unlikely to hold for a random ''a'' if ''p'' is composite.<ref name="PSW">{{cite journal |author1 = Carl Pomerance |author-link1 = Carl Pomerance |author2 = John L. Selfridge |author-link2 = John L. Selfridge |author3 = Samuel S. Wagstaff, Jr. |author-link3 = Samuel S. Wagstaff, Jr. |title=The pseudoprimes to 25·10<sup>9</sup> |journal=Mathematics of Computation |date=July 1980 |volume=35 |issue=151 |pages=1003–1026 |url=//math.dartmouth.edu/~carlp/PDF/paper25.pdf |jstor=2006210 |doi=10.1090/S0025-5718-1980-0572872-7 |doi-access=free }}</ref> Therefore, if the equality does hold for one or more values of ''a'', then we say that ''p'' is [[probable prime|probably prime]].

However, note that the above congruence holds trivially for <math>a \equiv 1 \pmod{p}</math>, because the congruence relation is [[Modular arithmetic#Basic properties|compatible with exponentiation]]. It also holds trivially for <math>a \equiv -1 \pmod{p}</math> if ''p'' is odd, for the same reason. That is why one usually chooses a random ''a'' in the interval <math>1 < a < p - 1</math>.

Any ''a'' such that 
:<math>a^{n-1} \equiv 1 \pmod{n}</math>
when ''n'' is composite is known as a ''Fermat liar''. In this case ''n'' is called [[Fermat pseudoprime]] to base ''a''.

If we do pick an ''a'' such that 
:<math>a^{n-1} \not\equiv 1 \pmod{n}</math>
then ''a'' is known as a ''Fermat witness'' for the compositeness of ''n''.