# What are some billions of digit prime numbers

## This huge new prime number is a very big thing

There is a new greatest known prime in the universe.

It's called M77232917 and it looks like this:

While it's a ridiculously large number (just the text file readers can download here takes up more than 23 megabytes of space on a computer), the M77232917 cannot be split without fractions. It doesn't break into whole numbers, no matter what other factors, big or small, someone divides it up. The only factors are yourself and the number 1. That makes it a top notch thing to do.

How large is these Number? A full 23,249,425 digits - almost a million digits longer than the previous record holder. If someone started writing it down 1,000 digits a day today (January 8th) it would end on September 19, 2081, as some calculations from WordsSideKick.com show on the back of the napkin.

Fortunately, there is an easier way to write the number: 2 ^ 77,232,917 minus 1. In other words, the new greatest known prime is less than 2 times 2 times 2 times 2 ... and so on 77,232,917 times. [The 9 most formidable numbers in the universe]

This is not really a surprise. Prime numbers that are one less than a power of two belong to a special class called Mersenne primes. The smallest Mersenne prime number is 3 because it is prime and also one less than 2 times 2. Seven is also a Mersenne prime number: 2 times 2 times 2 minus 1. The next Mersenne prime number is 31 - or 2 ^ 5- 1.

This Mersenne premiere, 2 77.232.917-1, surfaced in late December 2017 on Great Internet Mersenne Primes Search (GIMPS), a huge collaborative project involving computers around the world. Jonathan Pace, a 51-year-old electrical engineer, Germantown-based Tennessee who was involved in GIMPS for 14 years, receives recognition for the discovery that popped up on his computer. According to the January 3 GIMPS announcement, four other GIMPS hunters using four different programs confirmed the prime number within six days.

Mersenne primes get their name from the French monk Marin Mersenne, as University of Tennessee mathematician Chris Caldwell explained on his website. Mersenne, who lived from 1588 to 1648, suggested that 2 ^ n-1 is a prime number when n is 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, and none Prime for all other numbers less than 257 (2 ^ 257-1).

This was pretty good nudge for a response from a monk who worked three and a half centuries before the dawn of modern primate solving software - and a great improvement over writers prior to 1536 who believed that 2 was any prime with the number of times minus multiplied by 1 would be prime. But it wasn't quite right.

Mersenne's largest number, 2 ^ 257-1 - also written as 231,584,178,474,632,390,847,141,970,017,375,815,706,539,969,331,281,128,078,915,168,015,826,259,279,871, is actually not a prime number. And he missed a few: 2 ^ 61-1, 2 ^ 89-1, and 2 ^ 107-1 - although the last two weren't discovered until the early 20th century. Nevertheless, 2 ^ n-1 prime numbers bear the name of the French monk.

These numbers are interesting for a couple of reasons, although not particularly useful. One important reason: every time someone discovers a Mersenne prime, they also discovers a perfect number. As Caldwell explained, a perfect number is a number that is the sum of all positive factors (except for itself).

The smallest perfect number is 6, which is perfect because 1 + 2 + 3 = 6 and 1, 2, and 3 are all positive factors of 6. The next is 28, which is 1 + 2 + 4 + 7 + 14. After that comes 494. Another perfect number doesn't appear until 8,128. As Caldwell noted, these have been known since "before the time of Christ" and have spiritual meaning in certain ancient cultures. [5 Seriously Insane Math Facts]

It turns out that 6 can also be written as 2 ^ (2-1) x (2 ^ 2-1), 28 as 2 ^ (3-1) x (2 ^ 3-1), 494 equal to 2 ^ (5-1) x (2 ^ 5-1) and 8.128 is also 2 ^ (7-1) x (2 ^ 7-1). See the second part of these expressions? These are all Mersenne primes.

Caldwell wrote that 18th century mathematician Leonhard Euler proves that two things are true:

1. "k is an even number if and only if it has the form 2n-1 (2n-1) and 2n-1 is a prime number."
2. "If 2n-1 is prime, then so is n."

This means that every time a new Mersenne prime appears, a new perfect number.

That goes for M77232917 as well, although its perfect number is very, very large. The perfect twin of the large prime number, according to GIMPS in its statement, corresponds to 2 ^ (77.232.917-1) x (2 ^ 77.232.917-1). The result is 46 million digits long:

(Interestingly, all known perfect numbers, including this number, are the same, but no mathematician has proven that an odd number couldn't exist. Caldwell wrote that this is one of the oldest unsolved puzzles in mathematics.)

How rare is this discovery?

M77232917 is a huge number, but it's only the 50th known Mersenne prime. It may not be the 50th Mersenne in numerical order; GIMPS has confirmed that there is no missing Mersennes between the 3rd and 45th Mersennes (2 ^ 37.156.667-1, discovered in 2008), but the known Mersennes 46 through 50 may have skipped some unknown, previously undiscovered Mersennes .

GIMPS is responsible for all 16 Mersennes discovered since it was founded in 1996. These primes are not necessarily "useful" unless someone has found a use for them. However, Caldwell's website argues that the fame of the discovery should be reason enough, even though GIMPS has announced that Pace will be priced at \$ 3,000 for its discovery. (If someone discovers a prime of 100 million digits, the Electronic Frontiers Foundation will be awarded a prize of \$ 150,000. The first 1 billion digit prime is worth \$ 250,000.)

In the long run, Caldwell said, the discovery of more primes could help mathematicians develop a deeper theory about when and why primes appear. At the moment they just don't know, and it's up to programs like GIMPS to search with pure computing power.

Originally published on WordsSideKick.com.