Yesterday, GIMPS found the 50th Mersenne prime, now the world’s largest known prime. It is 23,249,425 digits long, so it’s really very big indeed.
GIMPS, the Great Internet Mersenne Prime Search, is a sort of distributed computing project which allows ordinary people to pool together to look for really big primes. This one, , was found by Jonathan Pace, a 51 year old electrical engineer. It took about 6 days and was independently verified by 4 other people and programs. Pace was awarded $3,000 for finding the prime, though it’s worth mentioning that he’s been looking for primes for 14 years now, so I imagine his electricity costs were not inconsiderable.
Mersenne primes are primes of the form . Mathematicians like looking for them because they have special properties which make them much easier to find than arbitrary prime numbers. For example, the exponent, p, must be prime. Secondly, the Lucas-Lehmer test (LLT), initially developed by Edouard Lucas in 1856, is an efficient way of testing whether a Mersenne number is prime. LLT is interesting because it flips the game around: it says a Mersenne number is prime if and only if it divides a particular larger number. So now all we have to do is perform long division once (admittedly on humongous numbers!).
In contrast, the largest known non-Mersenne prime has (only) 9,383,761 digits. It was found in 2016, but we’d already found Mersenne primes bigger than it in 2006, so it benefited from massively increased computing capabilities. The next biggest non-Mersenne prime has around 4 million digits. Clearly LLT is pretty good!
Finding a new Mersenne prime gives us a cool bonus: a new perfect number. A number is perfect if the sum of all its divisors is twice the number itself. For example, 6 is perfect since 1 + 2 + 3 + 6 = 12.
In the 4th century BC (!) Euclid proved that if is prime, then is a perfect number. In fact, it turns out that every even perfect number is of this form.
What about odd perfect numbers?
We’ve never found any odd perfect numbers, but we know that if one exists it must satisfy some fairly stringent properties. It must have over 1500 digits. It must have at least 101 prime factors. If you divide it by 324, you must get a remainder of 81, and if you divide it by 468, you must get a remainder of 117. There are a bunch more equally ridiculous conditions. In comparison, there are a fair few small even perfect numbers, like 6 and 28.
In 1888, James Joseph Sylvester said: “a prolonged meditation on the subject has satisfied me that the existence of any one such odd perfect number – its escape, so to say, from the complex web of conditions which hem it in on all sides – would be little short of a miracle.”