Fermat's Little Theorem - Art of Problem Solving
Fermat's Little Theorem is highly useful in number theory for simplifying the computation of exponents in modular arithmetic (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to Pierre de Fermat.
Fermat's Two Squares Theorem - Art of Problem Solving
Fermat's Two Squares Theorem states that that a prime number can be represented as a sum of two nonzero squares if and only if or ; and that this representation is unique. Fermat first listed this theorem in 1640, but listed it without proof, as was usual for him.
Fermat's Last Theorem - Art of Problem Solving
Fermat's Last Theorem is a recently proven theorem stating that for positive integers with , there are no solutions to the equation . History. Fermat's Last Theorem was proposed by Pierre de Fermat in the in the margin of his copy of the book Arithmetica.
Euler's Totient Theorem - Art of Problem Solving
Theorem. Let be Euler's totient function. If is a positive integer, is the number of integers in the range which are relatively prime to . If is an integer and is a positive integer relatively prime to , then . Credit. This theorem is credited to Leonhard Euler. It is a generalization of Fermat's Little Theorem, which specifies it when is prime ...
Art of Problem Solving
By Fermat's Little Theorem, we know that if . Therefore for all we have . Since , and is divisible by , of the possible are divisible by . Therefore, with probability or . Solution 2. Note that the patterns for the units digits repeat, so in a sense we only need to find the patterns for the digits .
Fermat's theorem - Art of Problem Solving
Pierre de Fermat invented famous theorems: Fermat's Last Theorem. Fermat's Little Theorem
Fermat’s Last Theorem - Art of Problem Solving
See https://artofproblemsolving.com/wiki/index.php/Fermat%27s_Last_Theorem. I don't know why this duplicate page was created.
Art of Problem Solving
Instead, take advantage of inverses in modular arithmetic. Just leave the negative powers of in a "storage base," and take care of the other terms first. Then, use Fermat's Little Theorem to solve for the power of . Video Solution: https://youtu.be/30CamkkifHM?t=766.
Chinese Remainder Theorem - Art of Problem Solving
Its ubiquity derives from the fact that many results can be easily proven mod (a power of a prime), and can then be generalized to mod using the Chinese Remainder Theorem. For instance, Fermat's Little Theorem may be generalized to the Fermat-Euler Theorem in this manner.
Art of Problem Solving
Category:Theorems. This is a collection of all theorems and provable formulas on the AoPSWiki.