# What has 17,425,170 digits, has sparked a positive flurry of excitement and has little practical value?

### The largest known prime number, of course - and it's like finding a diamond

A Missouri professor has discovered the largest prime number known to man, a 17-million digit integer which dwarfs the previous record-holder.

Mathematicians gazed in awe at the number, which is 17,425,170 digits long and most easily expressed as 2 raised to the 57,885,161 power minus 1.

The number would stretch 30 miles if printed in standard font. It can be downloaded in its entirety but that requires a 22.45 megabyte text file.

However the birth of 257885161-1, discovered by Curtis Cooper, a professor at the University of Central Missouri using hundreds of computers networked on the campus, provoked a debate within the scientific community.

Ever since Euclid found that there must be an infinite number of primes - numbers that can only be divided by themselves and 1 – mathematicians have competed to discover the largest statistical incarnation.

Almost 100,000 volunteers have joined Cooper in the Great Internet Mersenne Prime Search (GIMPS) project, tapping into a network of 730,562 processors, performing about 129 trillion calculations per second.

While the competition to uncover ever larger primes may produce an intellectual, and even aesthetic thrill for its practitioners, its real-life applications are limited.

Prime numbers inform the algorithmic encryptions used to keep web commerce transactions secure. But a 22 Mb number would be too unwieldy to have much practical use.

“It’s like finding a diamond,” Chris Caldwell at the University of Tennessee at Martin, who keeps a record of the largest known primes, told New Scientist. “For some reason people decide they like diamonds and so they have a value. People like these large primes and so they also have a value.”

The latest “diamond” is the 48th “Mersenne prime” to be discovered and eclipses the previous largest figure, a mere 12,978,189 digits, which was found by UCLA in 2008.

It took 39 days of non-stop computing on one of the University of Central Missouri’s PCs to verify Cooper’s result.

In 1640, French monk Marin Mersenne came up with a formula of 2p-1, with p being itself a prime number, which he said would produce large primes. Although it doesn’t work with every number, it gives a starting point for finding large primes and the GIMPS project tests likely candidates for Mersenne’s formula.

Cooper, who previously found record primes in 2005 and 2006, is one of 98,980 people and 574 teams involved in the GIMPS project, which has cornered the market in primes, discovering the last 11 largest numbers through its giant network of computers devoted to number-crunching.

A less cerebral attraction for prime number seekers is a cash reward. The Electronic Frontier Foundation will give \$150,000 to the discoverer of the first prime number with at least 100 million digits. A \$250,000 prize is on offer for the motherlode - a billion-digit prime. Cooper received \$3,000 for his latest discovery.

Prime number observers defended their latest find. A Fox News report claiming that the new prime had “little mathematical value” was condemned on Twitter.

However the online magazine Slate encouraged readers to be excited by the discovery. “Just finding one large prime number is a fun puzzle to have solved, but it doesn’t say anything basic about how the world works,” wrote Konstantin Kakaes.  “The patterns behind the primes, however, both proven patterns and ones only suspected, are the lens through which humanity can apprehend deep and unfamiliar truths about how reality is structured.”

George Woltman, a retired scientist who created the GIMPS network, argued that finding a new prime number was “analogous to climbing Mt Everest.” He said: “People enjoy it for the challenge of the discovery of finding something that's never been known before.”

History of Prime Numbers

Ancient Egyptians stumble upon the prime number principle with the Rhynd mathematical papyrus, four thousand years ago, through their calculation of unit fractions.

It took the ancient Greeks to demonstrate the existence of an infinite number of primes thanks to Euclid’s Theorem. 2,3,5,7,11…Prime numbers can only be divided by themselves and 1.

For every finite list of prime numbers, there is a prime number not on the list. Euclid concluded that there must then be infinitely many prime numbers, lighting the touch-paper on the race to find ever higher primes.

Euclid also proved the relationship between perfect numbers and Mersenne primes, a prime number which is one less than a power of two, later named after the French monk Marin Mersenne who studied them in the early 17th century.

Mersenne’s produced a formula of 2p-1 with p being itself a prime number, which would result in a larger prime. Mersenne primes are rare but the 10 largest-known primes are Mersenne’s.

Proving Pierre de Fermat’s Last Theorem, devised in the 1630s, eluded the mathematical community’s smartest minds for 350 years.  He died in the belief that he had found a relation which every prime number must satisfy.

Euler, Legendre and Gauss challenged and built upon Fermat’s Theorem. By the end of the 19th century the study of prime numbers threw out insights into how integers behave but was heading into abstract, number theory.

Relief for mathematicians as by the 1950s computer programmes are used to calculate those all-important largest-known Mersenne Primes.  The Great Internet Mersenne Prime Search (GIMPS), set up in 1996 using large-scale computer distribution, has calculated the 11 most recently-discovered largest-known primes.

Prime numbers found a real world utility from the 1970s in the algorithms used in credit card and, later, web commerce security. The product of the two prime numbers can be used as a “public key”, while the primes themselves are stored as a private key. Even the most powerful computers struggle to work out the factors of a large prime number.

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