Photo credit: Tucker Nichols

By Kenneth Chang

The largest known prime number, newly discovered, is almost five million digits longer than the previous record-holder.

In a computer laboratory at a satellite campus of the University of Central Missouri, an otherwise nondescript desktop computer, machine No. 5 in Room 143, multiplied 74,207,281 twos together and subtracted 1. It then checked that this number was not divisible by any positive integer except 1 and itself — the definition of a prime number.

This immense number can only be practically written down in mathematical notation using exponents: 2^{74,207,281} − 1.

The previous largest was 2^{57,885,161} − 1, which has a mere 17 million or so digits.

This is the 15th prime number found by the Great Internet Mersenne Prime Search, or Gimps, for short, a volunteer project that has been running for 20 years. “I’ve always been interested in prime numbers,” said George Woltman, who founded Gimps after he had retired. “I had a lot of time on my hands,” he said.

Continue reading by clicking the **name** of the source below.

## 5 comments on “New Biggest Prime Number = 2 to the 74 Mil … Uh, It’s Big”

you need large primes for encryption. Is there a formula that estimates what the odds any given number is prime? Do they get denser or sparser as N increases? Is there a fast algorithm for determining if any given number is prime, or is very probably prime?

Report abuse

RSA encryption uses a test for “probably prime”, because a complete test is too slow. As I recall, the complete test amounts to trying to divide by every prime less than the square root of the number being tested. With some shortcuts, but that’s basically it and it’s quite slow.

Report abuse

ROEDY

As far as I know, there is no discoverable pattern for the occurrence of prime numbers. Just consider the first century, 3 in the 10s, 4 in the teens, two in the twenties, two in the thirties, three in the 40s and 50s, two in the 60s, three in the 70s, two in the 80s, one in the 90s. I suppose that that means there could be a century with next to no prime numbers, and its neighbours could have loads of them. Russell’s antinomy was based on there being an infinite number of prime numbers.

I don’t know if it would be possible to get odds on a number’s being prime, since to do so it would be necessary to know the total occurrence of numbers and also of prime numbers. Since both collections are infinite, that’s a non starter, and since the occurrence of prime numbers is not predictable, the odds could not be calculated using an incomplete set of numbers. Bit like dog racing really.

Report abuse

The fraction of positive integers less than N that are prime is approximately log N.

There are a few primality tests with various pros and cons, and they mostly only show something is a probable prime. Mersenne numbers’ primality is easier to test (indeed, to establish with certainty), but such numbers are best avoided with RSA because you don’t want any “good guesses” for what your primes are.

Besides, for computational reasons RSA uses much prime numbers with at most a few thousand digits. Finding probable prime numbers that “small” with the full toolbox isn’t as hard as seeing whether a Mersenne number with many millions of digits is prime.

Report abuse

The total occurrence of numbers and also prime numbers is known. They are infinite, just like you said. It can actually be very easy to put odds on a random number having a certain property, even if the occurrence of that property is infinite. The odds on a random number to be even for instance.

Report abuse