1. (Mathematics) Fermat's
Little Theorem: Let "a" be a natural number (i.e. a number that is at least 1 and does not contain fractional parts), and let "p" be a (positive) prime number, where the greatest common divisor of "a" and "p" (i.e. gcd (a, p)) is 1, then, Fermat's
Little Theorem states that a ^ (p - 1) is equivalent to 1 (
mod p)
read: "a raised to the power of (p - 1) divided by p has a remainder of 1". Therefore, a^p is equivalent to a (
mod p) This is often used in RSA and other Public
Key Infrastructure.
2. (Mathematics) Fermat's Last Theorem: Let "x", "y" and "z" be non-zero integers, and let "n" be a natural number (
see above) greater than 2, then, there will not exist an equation x^n + y^n = z^n read: "(x to the power of n) plus (y to the power of n) equals (z to the power of n)" for any "x", "y", "z" and "n" combinations.
FLT
(Fermat's Little Theorem) Let "a = 4" and "p = 7", since 7 is
prime and "gcd (4, 7) = 1", then, 4^6 is equivalent to 1 (
mod 7) --> 4 to the power of 6 is 4096, and it is equal to 585 * 7 + 1. Therefore, 4 to the 7 is equivalent to 4 (
mod 7) i.e. 4 to the 7 is 16384, and it is equal to 2340 * 7 + 4.
(Fermat's Last Theorem) since this theorem asserts something does not exist, it is easy to come up with
random things, 3^5 + 4^5 =/= 5^5 --> (243 + 1024 = 1267 =/= 3125) where "=/=" means "not equal to".
