Advertisement

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 2

^{n}+ 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 2

^{n}+ 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^{n }+ 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 × 2^{n}+ 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 ×
2

^{31172165}+ 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," Prime Grid 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:

## Post A Comment:

## 0 comments: