New prime record: 51st Mersenne prime discovered

A new prime record was announced yesterday. The largest known prime is now

2^{82,589,933} - 1

Written in hexadecimal the newly discovered prime is

1\underbrace{\mbox{FFF \ldots FFF}}_\mbox{{\normalsize 20,647,483 F's}}

For decades the largest known prime has been a Mersenne prime because there’s an efficient test for checking whether a Mersenne number is prime. I explain the test here.

There are now 51 known Mersenne primes. There may be infinitely many Mersenne primes, but this hasn’t been proven. Also, the newly discovered Mersenne prime is the 51st known Mersenne prime, but it may not be the 51st Mersenne prime, i.e. there may be undiscovered Mersenne primes hiding between the last few that have been discovered.

Three weeks ago I wrote a post graphing the trend in Mersenne primes. Here’s the updated post. Mersenne primes have the form 2p -1 where p is a prime, and this graph plots the values of p on a log scale. See the earlier post for details.

Trend in Mersenne primes

Because there’s a one-to-one correspondence between Mersenne primes and even perfect numbers, the new discovery means there is also a new perfect number. M is a Mersenne prime if and only if M(M + 1)/2 is an even perfect number. This is also the Mth triangular number.

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