A few days ago, the discovery of a new record largest known prime number (282,589,933−1) was announced, making my post on “finding” record prime numbers somewhat obsolete. I could update it, but that would be silly. Instead, I’ll discuss these record primes.
It happens every couple of years: A new record large prime number is found, and some popular science articles are written about it. Then, on internet forums, you see questions like “Why are mathematicians so interested in finding prime numbers? What are prime numbers useful for?” These questions are based on some misunderstandings, perhaps helped along by journalists who oversimplify things to the point of being misleading or wrong.
When it is stated that 282,589,933−1 is the largest known prime number, it means that it is the largest number that has been proven (and generally known to humanity) to be prime. Only that and nothing more.
It does not mean that all prime numbers smaller than that number are known (they’re not). It does not mean not mean that mathematicians have a list of all known prime numbers (they don’t). It does not mean that finding a new prime number is difficult (it’s not). And it does not mean that prime numbers have some practical use (by and large, they do not).
There are way more smallish, easily discovered prime numbers than there are [insert your favorite big number metaphor here: atoms in the universe, whatever]. Making a list of them is physically impossible.
Sometimes a question is asked that boils down to “What is the smallest number for which we do not know whether it is prime?” This question does not have an answer, at least not until you define very precisely what it means to “know” whether a number is prime. Remember that there is no list of known prime numbers. We can, however, put some very broad bounds on the magnitude of such a hypothetical number. For instance, I can safely say that it is between 109 and 10100,000.
Will the search for large prime numbers put food on someone’s table? Cure a disease? Compose a sonnet? No, it won’t do those things. All it will do is add to humanity’s knowledge about numbers. I always find it strange that so many people do not consider the creation of new knowledge sufficient reason to do something.
Prime numbers do have uses in many computer algorithms, but the uses are mostly of the form “Algorithm A works best when parameter P is a prime number”, not of the form “If we could find a prime number, we could produce a fantastic new algorithm.”
And I suspect it’s even more limited than that. It seems like most every use of prime numbers is some variation on the same theme: They help to mix things up, create pseudo-randomness, eliminate unwanted resonances.
Another source of confusion is conflation of prime numbers in general with Mersenne primes, a specific class of prime numbers.
The Mersenne numbers are the numbers that are one less than a power of 2: 1, 3, 7, 15, 31, 63, 127, …. The Mersenne primes are the Mersenne numbers that are prime: 3, 7, 31, 127, ….
There are currently 51 known Mersenne primes, and the discovery of a new one is remarkable in and of itself. As a bonus, there’s a good chance that such a discovery will also represent the largest known prime number of any kind. That’s because Mersenne numbers are relatively easy to test for primality. So if you’re going to search for record large prime numbers, you might as well search for Mersenne primes.