51st Known Mersenne Prime Number Found (mersenne.org)

chalsall (Slashdot reader #185), writes: The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 2^82,589,933-1, having 24,862,048 digits. A computer volunteered by Patrick Laroche from Ocala, Florida made the find on December 7, 2018.

GIMPS has been on amazing lucky streak, finding triple the expected number of new Mersenne primes -- a dozen in the last fifteen years.
"This anomaly is not necessarily evidence that existing theories on the distribution of Mersenne primes is incorrect," notes GIMPS. "However, if the trend continues it may be worth further investigation. " They also report that the newly-discovered prime number "is more than one and a half million digits larger than the previous record prime number" -- and it's one of just 51 known Mersenne prime numbers ever discovered. "GIMPS, founded in 1996, has discovered the last 17..."

Patrick Laroche is one of thousands of volunteers using GIMPS' free software to hunt for prime numbers -- and is now eligible for a $3,000 "research discovery award," the group writes at mersenne.org. "GIMPS' next major goal is to win the $150,000 award administered by the Electronic Frontier Foundation offered for finding a 100 million digit prime number" -- of which $50,000 will be awarded to the discoverer, with another $50,000 going to a 501(c)(3) mathematics-related charity selected by GIMPS, and $50,000 retained by GIMPS to cover expenses and fund other awards.

Comments

No use for for cryptography but still interesting

[JoshuaZ]

And before anyone asks, no these large Mersenne are much too large to be used in practical cryptography. There is a random number generator called a Mersenne twister which does use a Mersenne prime, but that uses much smaller ones to be feasible, and in any event is not sufficiently random to be safe for serious cryptographic purposes.

The primary interest in these primes is two-fold: First they have a very efficient primality test, the Lucas-Lehmer test https://en.wikipedia.org/wiki/Lucas-Lehmer_primality_test and so if one is interested in simply finding very big primes, these are the ones to look for. For most of the last 100 years the largest known prime has beena Mersenne prime.

Second, there's a connection with perfect numbers. A number is said to be perfect if the sum of all its positive divisors which are less than the number add up to the number. For example, 6 is perfect because 1,2 and 3 divide 6 and 1+2+3=6. But 8 is not perfect because 1+2+4=7 which is not perfect. The two oldest unsolved problems in all of math are a) are there any odd numbers which are perfect and b) are there infinitely many even numbers which are perfect? About 2000 years ago, Euclid recorded a proof (which may or may not have been due to him) that every Mersenne prime allows you to construct an even perfect number. In the 1700s, Euler proved that any even perfect number must arise from Euler's construction. So if one cares about answering this question about even perfect numbers, then one wants to investigate Mersenne primes.

Re: To What End

[ledow]

Sigh.

You can get past a googol (10^100) with factorials of a 3 digit number. Factorials are thus "using" such numbers. Calculate the odds of N things chosen out of M and the numbers explode quickly.

Just because a number isn't representative of a practical physical quantity doesn't mean it's useless. Want an example? Encryption. I doubt you will ever exprienve having 2^4096 "anythings"... but the fact such a number exists, can be proven to be prime, and for which the mathematics applies is still incredibly useful.