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".
