Is the largest prime number known? If yes then show. If no then prove.

Question

I am not sure whether is known or not, I supposed not but don'nt know how to prove.

The largest prime number is not known, nor can it be known, for there are an infinite number of primes. No infinite subset of $\mathbb{N}$ possesses a maximal element. — MathematicsStudent1122, Jun 01 '16 at 23:22
That's a weirdly worded question. If there were a largest prime and we didn't know it, how on earth would we prove it wasn't known? — fleablood, Jun 01 '16 at 23:29

score 2 · Answer 1 · answered Jun 01 '16 at 23:24

There is no largest prime, as was shown in Euclid's Elements.

If you have a finite list of primes $p_1, \ldots, p_n,$ then the number $p_1 p_2 \ldots p_n+1$ is not divisible by any of those primes, so it must be a new prime or divisible by a new prime. So there must be infinitely many primes, and therefore no largest prime.

score 0 · Answer 2 · answered Jun 01 '16 at 23:20

0

The largest known prime number is $2^{74,207,281} − 1$. It is a Mersene prime, and it was discovered in January.

answered Jun 01 '16 at 23:20

Doug M

57,877

6

But there are bigger (as yet unknown) primes – Henry Jun 01 '16 at 23:24
I didn't mean to suggest otherwise. – Doug M Jun 01 '16 at 23:40
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This doesnot answer the question. – AaronS Jun 01 '16 at 23:59

Is the largest prime number known? If yes then show. If no then prove.

2 Answers2