In the last two months, two new Mersenne Primes have been discovered. Mersenne Primes are prime numbers that are 1 less than a power of 2. So for example, 7 is a Mersenne prime. These two new primes 243,112,609-1 and 237,156,667-1 are the 45th and 46th Mersenne primes known and are the two largest primes known. Their discovery has made it into the popular press and has prompted multiple people to ask me what the big deal is and why mathematicians care about big primes anyways. This blog entry will be the first of three entries in which I will attempt to discuss these issues.
First, note that there are infinitely many prime numbers. This has been known since the ancient Greeks. An elegant proof is given by
Unlike with generic primes, no one knows if there are infinitely many Mersenne primes. It is strongly believed that there are infinitely many but there is no proof as of yet. Mersenne primes are independently interesting for a variety of reasons. First, there is the issue of whether or not there are infinitely many. Second, Mersenne primes correspond to even perfect numbers. A number is said to be perfect if you add up all the positive divisors less than the number and end up with the number. So for example 6=1+2+3 so 6 is perfect.
Third, Mersenne primes are interesting because there are tricks to test whether a number of the form 2^n-1 is prime that make it much easier to check whether such numbers are prime than a generic number. Thus, for most of the last 100 years the largest primes known have been Mersenne primes.
Now, why do we care about finding large primes? One simple reason is for the same reason that scientists might try to construct larger and larger atoms or might look for new species; because they are there, learning is fun, and you don't know what you might find.
Also, in a more general setting primes (although generally primes much smaller than Mersenne primes) are useful for cryptography which are used in many modern software applications. Many readers have likely used those cryptographic protocols without even realizing it when you've logged onto to secure websites and such.
Modern browsers handle the procedures for you automatically. So the ability to find large primes and related procedures (such as theability to factor large numbers quickly) are very relevant to the functioning of the modern economy as well as to various national security issues. I will discuss this cryptographic use of large primes in the third blog entry in this series.