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I want to know, what is the largest prime number? I know prime numbers are whole numbers that cannot be divided by any whole number except 1 and themselves, I also know some primes like 2, 3, 5, 7, 11... But what is the largest prime number?
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There is no largest prime number, since there are infinitely many prime numbers as proved by Euclid back to 300BC. However, there is a largest known prime which is: $$2^{57885161}-1 $$ as of January 2014, discovered by the GIMPS project. You can even help find a new one, by following the simple steps described here and letting your computer do the job.
When you try to catch the latest prime number:
(づ｡◕‿‿◕｡)づ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 53$
Another one appears behind you
$59 \ \ \ \ \ \ \ \ \ \ \ \ $ (づ｡◕‿‿◕｡)づ $\ \ \ \ \ \ \ \ \ \ \ \ 53$
They're infinite!
Euler proved it with the zeta function:
$$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p}^\infty \frac{1}{(1-\frac{1}{p^s})}\tag{$p$ is a prime}$$
I would like to provide a little more detail and offer the proof for the infinitude of the primes. Suppose that there was a last prime number. If we numbered the primes, we'd have $p_1$, $p_2$ and so on until the last one which we'll call $p_k$.
We now consider the product of all of these prime numbers plus 1. $n=p_1p_2p_3\cdots p_k+1$. To check to see if a number is prime, it suffices to divide by all prime numbers smaller than it, so if we divide out any of the primes, maybe $p_i$ for all $i$ between $1$ and $k$, we would have then ${p_1p_2\cdots p_{i-1}p_ip_{i+1}\cdots p_k$+1} \over p_{i}$. We could simplify this term then to include a product of primes on the left, cancelling out $p_i$, but then we would still have the last term $1\over{p_i}$, which is not an integer since $p_i$ is greater than 1. But that means that $n$ was not divisible by $p_i$. Since $i$ was any arbitrary positive integer less than or equal to $k$, $n$ is not divisible by any of the primes smaller than itself, so $n$ too is prime, contradicting the idea of a last prime.
So we conclude their are infinitely many primes.
The largest prime we know till now, is $2^{57885161}-1$
source: http://en.m.wikipedia.org/wiki/Largest_known_prime_number
There is no biggest prime. Its just that we currently don't know what is the next biggest prime, but there is one!
