## A new Mersenne prime confirmed today! Well, sort of . . . .

Saw this tweet from Chris Long today:

and then the sensational followup from Aperiodical:

So, what happened?  After about 8 years of computation the folks at the Great Internet Mersenne Prime Search ( ) announced today that the prime number $2^{32582657} - 1$ has been confirmed as the 44th Mersenne prime.    What’s particularly interesting about this announcement is that this prime number itself was found in 2006 – the extra 8 years were spent confirming that there were no other unknown Mersenne prime numbers lower than that number.   Quite an amazing project.

So, clever tweets aside, a fun discovery and a great opportunity to talk about some current (as in *today* current) math with the kids tonight.

We started by just discussing a little bit of history of Mersenne primes and calculating a few small ones.  My younger son remembered a bit about Mersenne primes from our Number Theory book which was nice.  One fun thing about the history of these numbers is that both Euler and Euclid played important roles in our understanding of these primes:

The next thing we did talk about one of the neat applications of Mersenne primes – perfect numbers.  The point here was to review perfect numbers a little and also see how to create perfect numbers from Euler’s formula.  It is also neat to be able to show the boys an unsolved problem in math that they can understand – are there any odd perfect numbers?  This part, btw, would be a fun project all by itself – it isn’t super hard to see why these number will be perfect numbers (seeing why they are the only even ones is probably a little harder, but I don’t know that proof so I don’t know how difficult it is).

Next we talked about the Great Internet Mersenne Prime Search and the progress that this program has made in our understanding of these primes.  It is a really neat project which, among other things, has found the largest known prime numbers.  We talk about some of the largest primes that they’ve found and how big those numbers are.   We also talk about the work that had to be done to determine that the number found in 2006 was indeed the 44th Mersenne prime:

Finally, a fun piece of math for the kids to work on – how many digits are in the decimal representation of the number $2^{32582657} - 1$?  Definitely a challenging problem, but one that the kids can make progress on with a little work.   The boys found what they thought was a pattern in the number of digits of powers of two and used that to take a guess at the number of digits of the large prime.  I showed them an alternate approach that uses the idea that $2^{10}$ is approximately 1,000.   We got estimates that were reasonably close to the right answer:

I really like the opportunity to share current developments in math with the kids.  It is extra nice when the developments are ones that they can understand and even play around with themselves.  There are some fun properties of  Mersenne primes that kids can play around with, and even some easy to understand unsolved problems, too.   Estimating the number of digits in these extremely large numbers is also an accessible exercise.  What better time to explore these numbers than on the day a neat discover was announced!

One Comment so far.
1. Dickson’s proof is very pretty.
Let N = 2^(k-1) * m be an even perfect number where m is odd.

σ(N) = σ(2^(k-1)) * σ(m)
2N = (2^k – 1) * σ(m)
2^k * m = (2^k – 1) * σ(m)
σ(m) = (2^k * m) / (2^k – 1)
σ(m) = m + m/(2^k – 1)

Let d = m/(2^k – 1). Then σ(m) = m + d, and d is a divisor of m.
But σ(m) is the sum of all divisors of m, including m and d.
Therefore m is prime and d = 1, so m = 2^k – 1.