I'm not sure if you're prepared well enough, but you could get advice and start your own search here
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GFN-14: 50103906^16384+1
Proth "SoB": 44243*2^440969+1

Here is a list of all known primes (or probable primes) of this form. As Kellen said, we currently have no practical way to prove that a really large number of this form is actually prime, hence why these are only "probable primes" beyond k=10988.

We can certainly use Fermat's little theorem--but Fermat's little theorem never tells us that something is guaranteed to be prime, only that if p is prime, then a^(p-1) is congruent to 1 mod p for all integers a not divisible by p. The converse is usually true, but not always, because of Carmichael numbers (e.g. 561) and other pseudoprimes (e.g. 341 in base a=2).

So if we test Fermat's congruence for a candidate p with several different choices of a, we can be highly confident, but not 100% confident, that it's prime. In fact, when we call something a "probable prime", that usually means that we've done these tests, perhaps along with some trial factoring and a few souped-up versions of the same idea (e.g. Miller-Rabin or BPSW) for extra confidence.

One more note: since (as you said) these tests involve really large numbers, we have to be slightly careful when implementing them. The number 2^(3^504207 + 2) - 1 has far more digits than there are atoms in the universe, so there is no hope of representing it in binary on any physical computer. But fortunately we don't need to: we only need its congruence class modulo the 240568-digit number p = 3^504207 + 2. We can do all our calculations mod p, and some modular exponentiation tricks make it possible to compute 2^(3^504207 + 2) - 1 mod p with only about a million multiplication steps. This is feasible on modern computers.

Just noticed this pop up on PrimePages: https://primes.utm.edu/primes/page.php?id=130986

Looks like someone is working to prove primality for these :)