What is the largest prime number?

Question

I have heard it once and I know that it is huge! Yeah, I guess it took four or five guys four whole months to find it out!

Answer 1

The set of prime numbers has an infinite number of elements. There is no largest prime number.

Proving that a particular number is prime, however, is an entirely different kettle of fish. That requires huge amounts of computing power. There is one group that uses distributed computing to do this (called GIMPS - link 1 below)... their most recent world record was set on September 4, 2006. There are tens of thousands of people participating in this, so it's not like just four guys get the credit.

For the record, that prime was two to the power of 32,582,657 minus one. It has 9,808,358 digits (link 2, if you want to see them all - be prepared to annoy your computer by looking at them all).

Hope that helps!

Answer 2

The largest *known* prime number is 2^32582657-1. It's also known as Mersenne 44, I think.

There is no largest prime number, however. By the Fundamental Theorem of Arithmetic, every positive integer can be factored into primes in exactly one way. Now, suppose there is a largest prime number. This means that there's a finite number of prime numbers, so we can multiply them all together to get some bigger number. Let's call that number n.

Now, n has every prime number as a factor. If we add 1 to any positive integer, it loses all its prime factors, so n+1 must have no prime factors. However, the only positive integer that has no prime factors is 1, so n must be 0. Multiplying positive integers will never get 0, as 0 is not considered positive, so 0 can't be the product of all prime numbers.

So we've proven that if there's a product of all prime numbers, it must be 0, but we've also proven that a product of prime numbers can't be zero. This means that the product of all prime numbers must not exist, meaning the number of prime numbers is infinite, meaning there is no largest prime number.

Answer 3

The largest KNOWN prime is a merseinne prime, as others have noted.
However, we knew as far back as the time of Euclid that the set of prime numbers, like the set of integers, is countably infinite. Here's some proofs..
http://primes.utm.edu/notes/proofs/infinite/
Although we know in theory that the set of primes is infinte, finding prime numbers gets increasingly labor intensive as the numbers get larger. Even though this labor is done by computer, it is still a great deal of work and energy to be spent. This, rather than some limit on the set of primes, is the reason why there is a largest KNOWN prime number even though there is no largest prime number.

Answer 4

Too large to state!

Quote
A team at Central Missouri State University (CMSU) broke its own world record last week by discovering the largest prime number to date. It has an impressive 9,808,358 digits, but can be conveniently written as (2 to the power of 32,582,657) -1. Unfortunately, though, it is not quite large enough to win its discoverers the $100,000 award offered by the Electronic Frontier Foundation for the discovery of the first 10 million digit prime number.


reference
http://plus.maths.org/latestnews/sep-dec06/mersenne/index.html

Answer 5

First, the largest number, period cannot be determined because, whatever number anyone can come up with, you can always add, at least one.

So, since the there is no "largest" number, as long as there is infinity, no one can determine the largest prime number.

Someone may have calculated the largest prime number, discovered so far, but no one can say that it is the absolute largest prime number possible.

Answer 6

So, let's be specific here. Just like there are an infinite number of odd numbers and an infinite number of even numbers and an infinite number of square numbers, there are an infinite number of prime numbers.

There is no one largest prime number, and this was proved a few thousand years ago by Euclid in a famous proof that you can read here: http://www.math.utah.edu/~pa/math/q2.html

This means that no matter how big a prime number we come up with, there's always a bigger one that we just haven't found yet. But coming up with prime numbers isn't easy. It takes a lot of computer time.

Here's the biggest prime number we've found SO FAR:
2^(13,466,917) - 1
You can see it written out longhand here:
http://www.math.utah.edu/~pa/math/largeprime.html

There are bigger prime numbers, but we haven't had time to compute what they are yet -- of course, someday we will!

Answer 7

There is no largest prime number. Whatever prime number you can name, a larger one can be found.

If x is the prime number you are thinking of, then x! + 1 is a greater prime number.

Answer 8

Big difference between largest prime (cannot be known, as numbers are infinite) and largest KNOWN prime (see answers above!)

Answer 9

The largest known prime is also a Mersenne prime: (2^6972593)-1

Answer 10

x! + 1 is always a prime number where x is an integer

there is no highest integer, so there is no highest prime number.