How, if at all, does pure mathematics benefit from $2^{74207281}-1$ being prime?

Question

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.

Answer 1

It has no practical applications in mathematics or any other field as far as I know.

It is expected that there are an infinite number of Mersenne primes, so discovering a new one is completely in line with that. On the other hand, it would be much more interesting and perhaps somehow consequential if we failed to find a new one where one ought to be according to the conjectured distribution. Or if we found two whose exponents were very close together, that would also be quite a surprise begging for a mathematical explanation; there is no way to rule this out currently, it may be the case that $2^p-1$ is prime for all sufficiently large prime $p$. But this one is completely ordinary-looking and its discovery is just one more piece of evidence for what is already widely-believed.

RSA cryptography relies on the difficulty of factoring composite numbers whose prime factors are unknown, so big Mersenne primes have no relationship to this. There are other cryptosystems based on the discrete logarithm problem such as Diffie-Hellman and ElGamal that use known prime numbers as parameters, but much smaller primes are secure enough, and Mersenne primes are considered unsuitable because they are not safe.

Mersenne primes are used in PRNGs like MT19937 because it is faster to perform multiplication in binary modulo a Mersenne prime versus other types of primes. But it will take some decades of continuation of Moore's law before giant Mersenne primes find a role here and that seems quite unlikely.

So why does anybody care then? Because it's an awesomely difficult accomplishment that inspires the public and gets people interested in learning more about mathematics. And what aspiring and ambitious mathematician wouldn't be proud to see their name preserved in history alongside Leonhard Euler? It's the type of thing that we celebrate for its own sake.

Answer 2

I made a list of reasons why mathematicians care about perfect numbers and Mersenne primes which may be relevant. But the search is mainly done for aesthetic reasons, not because of those applications. However, there are also spillover benefits. The search for Mersenne primes led to the discovery of a flaw in certain Intel processors, for example. More importantly, it has led the development of fast FFT software for large numbers, which have myriad applications in industry. (The search uses relatively large numbers, so they have to develop new software to run efficiently there, but as computers become generally more efficient those sizes become useful for many practical projects.)

But I can't think of any reason that knowing that this particular number is prime would advance mathematics, not on its own. If enough Mersenne primes are found, though, it will provide evidence for or against standard conjectures about how they are distributed. If it's what we expect that changes very little, but if they aren't then finding out why and how are suddenly very important questions, because that might undermine other assumptions built on apparently-faulty premises.

Answer 3

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):

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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...