Most the of list of large prime numbers are Mersenne primes, those of the form 2^n -1. It turns out that if 2^n -1 is prime, then n is prime,
so that limits the cases that need to be checked: just look at large primes n and consider 2^n -1.
https://en.wikipedia.org/wiki/Mersenne_prime
Testing divisibility by all smaller primes (or even those up to the square root) is a horrible way to test primality of large numbers. Even with fast, distributed computers on every atom in the observable universe, it would literally take man times the age of the universe to do it this way.
There is a more convenient way to show that some number is NOT prime using what is known as Fermat's little theorem (not to be confused with his last theorem):
If p is prime and 1<a<p, then the remainder of a^(p-1) when divided by p is 1.
This test is often used to get potential prime numbers for security operations (Mersenne primes would never be used for such).
The reason most known large primes are Mersenne primes is that there is a fast primality test for such, called the Lucas-Lehmer test.
https://en.wikipedia.org/wiki/Lucas%E2%8...ality_test
This still takes a lot of computational power and testing numbers like the one for this thread is still done by distributed computing, but it is *possible*.
Any questions?
so that limits the cases that need to be checked: just look at large primes n and consider 2^n -1.
https://en.wikipedia.org/wiki/Mersenne_prime
Testing divisibility by all smaller primes (or even those up to the square root) is a horrible way to test primality of large numbers. Even with fast, distributed computers on every atom in the observable universe, it would literally take man times the age of the universe to do it this way.
There is a more convenient way to show that some number is NOT prime using what is known as Fermat's little theorem (not to be confused with his last theorem):
If p is prime and 1<a<p, then the remainder of a^(p-1) when divided by p is 1.
This test is often used to get potential prime numbers for security operations (Mersenne primes would never be used for such).
The reason most known large primes are Mersenne primes is that there is a fast primality test for such, called the Lucas-Lehmer test.
https://en.wikipedia.org/wiki/Lucas%E2%8...ality_test
This still takes a lot of computational power and testing numbers like the one for this thread is still done by distributed computing, but it is *possible*.
Any questions?
