# What Is The Largest Prime Number?

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## 8 Answers

There are an infinite number of primes so there isn’t a largest prime. I have a truly marvellous demonstration of this proposition which this margin is unfortunately too narrow to contain.

The largest **known** prime number (as of November 2018) is 2^77,232,917 − 1, a number with 23,249,425 digits. It was found by the Great Internet Mersenne Prime Search (GIMPS) in 2017.

There is no largest prime, and there is a really simple proof due to Euclid.

We will show that for any number n, there is a prime p, with p > n. Choose any number n. There are k primes less than or equal to n for some value k. Multiply them all together and then add 1 to get x = 2*3*5*...*pk + 1. x is not divisible by any of the primes, since it has a remainder of 1 when you divide x by any of them.

There are two possibilities. x may be a prime, in which case it must be greater than n, since it would be different from any of the primes less than or equal to n. Set p=x. On the other hand, x may not be a prime, in which case it can be expressed as the product of primes. All of these prime factors must be greater than n, since x is not divisible by any prime less than or equal to n. Set p equal to any of these prime factors.

We can choose n to be arbitrarily large, which means that the primes greater than n are arbitrarily large, meaning there is no largest prime.

As an example, find a prime > n=6. The primes <= 6 are 2, 3 and 5. x = 2*3*5+1 = 31. Since 31 is a prime, so we can set p = 31.

Not to confuse things, but an interesting aspect of Euclid’s proof is that it can be used to generate an arbitrarily large sequence of numbers that are not prime.

Using the terminology from the proof, let y = 2*3*5*...*pk. We know that y+1 may be prime, but the numbers y+2, y+3, ... y+n are all composite (non-prime). Consider any y+z, with z between 2 and n. Since z <= n, it must be divisible by one of the primes <= n. Call the prime p. We know that y is divisible by p. Therefore y+z is divisible by p and not equal to p. so it must be composite.

Using the example of primes < 6, we have 2*3*5=30, and 32, 33, 34, 35,and 36 are all composite. We can’t go any further in this case, since 37 is a prime.

Let ∞/2 = k

2k = ∞

Twice infinity is infinity therefore k equals infinity

Let ∞/3 = n

3n = ∞

Thrice infinity is infinity therefore n equals infinity

All primes numbers larger than 2 or 3 are either 1 more than or 1 less than a number divisible by 6

Ergo the largest prime number is infinity plus or minus 1

^^^^What they said. But minus 1.

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