algorithm/JavaScript
1
0
Fork 0
mirror of https://github.com/TheAlgorithms/JavaScript.git synced 2025-07-05 00:01:37 +08:00
Code Issues Packages Projects Releases Wiki Activity

Minor fixes

Browse Source 
- Add some explanation and links about Carmichael Numbers
- Remove explanation about in-built function Math.random()
This commit is contained in:
arthurvergacas
2021-10-16 11:37:15 -03:00
parent ffb138adb1
commit fa1abf4e1c
1 changed files with 23 additions and 9 deletions
Download Patch File Download Diff File Expand all files Collapse all files

32
Maths/FermatPrimalityTest.js
View File

@ -1,26 +1,40 @@
/*
* The Fermat primality test is a probabilistic test to determine whether a number is a probable prime.
* The Fermat primality test is a probabilistic test to determine whether a number
* is a probable prime.
*
* It relies on Fermat's Little Theorem, which states that if p is prime and a is not divisible by p, then
* It relies on Fermat's Little Theorem, which states that if p is prime and a
* is not divisible by p, then
*
* a^(p - 1) % p = 1
*
* However, there are certain numbers (so called Fermat Liars) that screw things up;
* if a is one of these liars the equation will hold even though p is composite.
*
* But not everything is lost! It's been proven that at least half of all integers aren't Fermat Liars (these ones called
* Fermat Witnesses). Thus, if we keep testing the primality with random integers, we can achieve higher reliability.
* But not everything is lost! It's been proven that at least half of all integers
* aren't Fermat Liars (these ones called Fermat Witnesses). Thus, if we keep
* testing the primality with random integers, we can achieve higher reliability.
*
* The interesting about all of this is that since half of all integers are Fermat Witnesses, the precision gets really
* high really fast! Suppose that we make the test 50 times: the chance of getting only Fermat Liars in all runs is
* The interesting about all of this is that since half of all integers are
* Fermat Witnesses, the precision gets really high really fast! Suppose that we
* make the test 50 times: the chance of getting only Fermat Liars in all runs is
*
* 1 / 2^50 = 8.8 * 10^-16 (a pretty small number)
*
* For comparison, the probability of a cosmic ray causing an error to your infalible program is around 1.4 * 10^-15. An
* order of magnitude below!
* For comparison, the probability of a cosmic ray causing an error to your
* infalible program is around 1.4 * 10^-15. An order of magnitude below!
*
* But because nothing is perfect, there's a major flaw to this algorithm, and
* the cause are the so called Carmichael Numbers. These are composite numbers n
* that hold the equality from Fermat's Little Theorem for every a < n (excluding
* is factors). In other words, if we are trying to determine if a Carmichael Number
* is prime or not, the chances of getting a wrong answer are pretty high! Because
* of that, the Fermat Primality Test is not used is serious applications. :(
*
* You can find more about the Fermat primality test and its flaws here:
* https://en.wikipedia.org/wiki/Fermat_primality_test
*
* And about Carmichael Numbers here:
* https://primes.utm.edu/glossary/xpage/CarmichaelNumber.html
*/
/**
@ -66,7 +80,7 @@ const fermatPrimeCheck = (n, numberOfIterations = 50) => {
if (n <= 3) return true // 2 and 3 are included here
for (let i = 0; i < numberOfIterations; i++) {
// pick a random number a, with 2 <= a < n - 2 (remember Math.random() range is [0, 1[ -> 1 exclusive)
// pick a random number a, with 2 <= a < n - 2
const randomNumber = Math.floor(Math.random() * (n - 2) + 2)
// if a^(n - 1) % n is different than 1, n is composite