system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5:Why is it important to keep finding big prime numbers ?
gpt
And, there are practical uses as well.  

 - There are mathematical techniques that turn prime numbers into locks.  The bigger the number, the stronger the lock.

 - Some of these locks make keys from reasonably large prime numbers chosen at random.

 - Numbers made of solid ones in binary are easy for computers to work with.  (11 111 1111 11111 111111 etc.)  These are called "Mersenne numbers", which makes Mersenne *primes* especially useful large primes.  They're easy to guess, so you can't use them as "random" large primes.  

Why not:

You might write some code that can do math with 4096-bit numbers.  

The smallest period over time for which modern physics makes sense is the Planck time.  If you start at the Big Bang and start counting the age of the universe using "Planck times" as your units, you would be up to 61-digit numbers by now.  In binary, that's 203 bits.

You cannot imagine how many 203-bit numbers there are without a powerful metaphor like that.  You would need a much more powerful metaphor to get from 203-bit numbers to 4096-bit numbers.

It's easy to pick random 4096-bit numbers, obviously, and math with them is decently fast.  Thus the usefulness in cryptography.  

Of the 4096-bit numbers, about 1 in 2839 are prime.  It's surprising that we can estimate such precise odds, but the Prime Number Theorem makes it possible.  Take the number of digits (4096), multiply the natural logarithm of the base (ln 2).  The PNT says that this estimate becomes more correct with more digits.

Mersenne numbers are much rarer.  So rare, you cannot imagine how rare they are.  In the near-infinity of 4096-bit numbers, only 4096 of them are Mersenne numbers.  Only some of the Mersenne numbers are prime.

You cannot guess a randomly-selected 4096-bit prime, even with an entire universe of computing power.  But it's pretty easy to check a mere four thousand possibilities.

You could knock that out with a calculator if you were bored.