The world's largest prime number to date :

Discovered less than 2 month ago by the Great Internet Mersenne Prime Search or GIMPS, an online distributed computing project, this is the largest prime number known to date.

It has more than 24 million digits (a large book).

Today's 7 largest prime numbers are Mersenne primes, meaning they're of the form 2^n -1. This is due to the fact that these numbers are easier to "test", meaning to check if it has any divisors.
For example if 2^n -1 is prime then n is also a prime number, this is an implication but still allows us to exclude certain numbers.
Many different number theory theorems are used to narrow the number of potential large numbers. This allows us to find these incredibly large primes.
The special algorithms designed to check if a given number is prime effectively brute force different factorization, there is no way to tell if a number is prime other than checking all of its divisors. Obviously they rule out certain numbers like evens, divisors by 3 and such.
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