TITLE: How, if at all, does pure mathematics benefit from $2^{74207281}-1$ being prime?
QUESTION [16 upvotes]: So a couple of days ago the $17$ million digit number $2^{57885161}-1$ was beaten by the $22$ million digit number $2^{74207281}-1$ at being the largest known prime number.
Are there any specific (purely mathematical) implications of the fact that this particular number is prime? In particular, does this resolve something other than the question if $2^{74207281}-1$ is prime, in the style of some newly discovered theorem resolving previous conjectures? Is it perhaps a counterexample to something so far believed to be true?
Edit: I'm aware of the many non-(strictly mathematical) practical implications and reasons to continue the search for large primes (cryptography etc), but I'm not aware of any strictly mathematical uses of particular numbers being prime, which is why (and what) I'm asking.

REPLY [2 votes]: Well, there have been cases in the past where a conjecture was eventually refuted only for an extremely large value. So finding such number (and at a certain distance from the previous number), may contribute to our knowledge of some yet unproven conjecture.
Here is one such example of a conjecture that was eventually refuted only for an extremely large value (though in this case it was not refuted as a direct result of discovering that value):

Consider the following functions:

The prime-counting function $\pi(n)$
The logarithm-integral function $Li(n)=\int\limits_{2}^{n}\frac{1}{\ln{x}}dx$

The prime number theorem, stating that $\pi(n) \approx Li(n)$, was proved in $1896$.
For small values of $n$, it had been checked and always found that $\pi(n)<Li(n)$.
As a result, many prominent mathematicians, including no less than both Gauss and Riemann, conjectured that the inequality was strict.
To everyone's surprise, this conjecture was refuted in $1914$.
It's worth noting that it was refuted not by a counterexample, but by proving that a counterexample existed (in fact, an exact counterexample remains unknown to this day).
Read this for more details...