Steve Johnson/Flickr

A new prime number has been found, and it's 9.3 million digits long

It could help us solve a decades-old puzzle.

BEC CREW
28 NOV 2016
 

Thousands of collaborators from all over the world have come together to find one of the largest known prime numbers, and the discovery has gotten us closer than ever to solving the decades-old Sierpinski problem.

At more than 9 million digits long, the new prime number is the seventh largest prime ever found, and it just cut the six possible candidates for the elusive Sierpinski number down to five.

 

Established by Polish mathematician Wacław Sierpiński in the 1960s, the Sierpinski problem asks you to find the smallest possible number that meets a specific - and very tricky - set of criteria. 

A Sierpinski number must be a positive, odd number, and takes the place of k in the formula k x 2n + 1, for which all integers are composite (or not prime).

In other words, if k is a Sierpinski number, all constituents of the formula k x 2n + 1 are composite.

The trick is that, in order to prove that k is a Sierpinski number, you have to show that k x 2+ 1 is composite for every n. If n = a prime number, you’re out of luck.

"This has to hold for any positive, whole value of n," says Timothy Revell at New Scientist. "These numbers are few and far between, so they aren’t easy to find."

Right now, the lowest known Sierpinski number is 78,557, proposed by American mathematician John Selfridge back in 1962, but how do we know there aren’t smaller ones?

 

Over the past 50 years, mathematicians have found six possible candidates that could be the smallest possible Sierpinski numbers: 10,223, 21,181, 22,699, 24,737, 55,459 and 67,607. But so far, no one’s been able to prove that any of them are definitely a Sierpinski number.

"To be certain you’re really dealing with a Sierpinski number requires a mathematical proof that no matter what choice of n you make, k × 2n + 1 will never end up prime," says Revell.

To know that, you have to know what numbers are prime numbers, and that’s where the PrimeGrid 'Seventeen or Bust' project comes in.

The project gets volunteers to help in the search of large prime numbers by allowing their computers to do the necessary calculations to prove that a certain number is a prime. 

"Users download software to their PC and then can join different groups depending on the type of prime numbers they are interested in looking for," Iain Bethune from PrimeGrid told New Scientist.

In an effort to solve the Sierpinski problem, the project has just found its largest prime number, and the seventh largest prime number on record: 10,223 × 231172165 + 1.

At 9,383,761 digits long, a single PC would have taken centuries to find this prime - this monster prime is the result of thousands of computers teaming up to find it over an eight-day period. 

But this new prime is special for another reason - it has disproven one of the six candidates for the Sierpinski number.

"This is the largest prime found attempting to solve the Sierpinski Problem and eliminates k = 10,223 as a possible Sierpinski number," PrimeGrid recently announced.

And then there were five.

Btw, if you think 9.3 million digits is impressive, the largest known prime number was discovered back in January, and it's a whopping 22 million digits long. 

Interestingly, that new record-breaker is part of a rare group of prime numbers known as Mersenne primes - which means that it can be written as one less than a power of two - but the new 9.3-million-digit prime is not. 

In fact, among the 10 largest known prime numbers, our new prime is the only prime that is not a Mersenne number, and the only known non-Mersenne prime over 4 million digits.

While finally solving the Sierpinski Problem will probably only be a big deal to mathematicians - and number fans like us - finding the largest primes are crucial for researchers to improve encryption technology and computer power.

Here's what a 22-million-digit number looks like on paper:

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