Just another facet of primes: Who dreamed up the primes knots analogy? and What is the knot associated to a prime?

One way to get big numbers quickly is to use exponent towers. These are prime:

PrimeQ[3^7^3+4^6^4]
PrimeQ[7^6^3+6^5^5]
PrimeQ[3^7^2+8^5^5]

To get a lot of large primes with small representations, visit the Probable Prime Records page. Virtually all known primes and probable primes with more than 100000 digits have a short, elegant representation. Why? Because it takes awhile to do the primality tests, and no-one is looking at big random primes.

Among actual primes, there are Primorial primes 1098133#-1 (476311 digits), Factorial primes 150209!+1 (712355 digits), Cullen primes 6679881?·?2^6679881 + 1 (2010852 digits), Proth primes 19249?·?2^13018586 + 1 (3918990 digits).

The probable primes are much more varied. Elliptic curves can be used to prove primality, but that only works up to about 22000 digits at the moment.

The Cunningham project factors numbers of the type a^b+1 and a^b-1. I recently put up a Demonstration that factors up to 2^1200 + 1 and 2^1200 - 1, for those where solutions are known.

It would be amazing if one could actually make this analogy computable,

... at least I learned recently there exists for a while already the Khovanov Homology, as a reference see On Khovanov's categorification of the Jones polynomial and, also by Bar-Natan: Khovanov Homology for Knots and Links with up to 11 Crossings the latter one could by used to check oneself's understanding.