[akalin]

You still mention that:

> There's no slick trick to check fast enough, whether or not a large number is prime. So much so that finding large prime numbers has been an obsession for mathematicians.

But that's still false; in fact, the rest of your article talks about the Fermat and Miller-Rabin primality test, which are indeed slick tricks that can check fast enough whether or not a large number is prime (to whatever degree of confidence you desire)!

Also, finding large prime numbers isn't really an obsession anymore -- you can use any of the fast primality test algorithms mentioned above, randomly generate numbers of a desired bit length, and stop when you hit a (probable) prime, which happens fairly quickly by the prime number theorem: https://en.wikipedia.org/wiki/Prime_number_theorem

You may be thinking of the search for Mersenne primes, or other primes of a specific form.