Infinite prime numbers

[WalkerDPlank]

On April 12th, the 47th known Mersenne prime, 242,643,801-1, a 12,837,064 digit number was found by Odd Magnar Strindmo from Melhus, Norway! This prime is the second largest known prime number, a "mere" 141,125 digits smaller than the Mersenne prime found last August.
Would it not be reasonable to think that there are higher prime numbers, in fact wouldn't it be reasonable to think that if you keep on going there would be an infinite amount of prime numbers?

 

[PhatoseAlpha]

The infinitude of primes was proven 23 centuries ago. So, you're kind of late on that discussion.

 

[TuxDave]

On April 12th, the 47th known Mersenne prime, 242,643,801-1, a 12,837,064 digit number was found by Odd Magnar Strindmo from Melhus, Norway! This prime is the second largest known prime number, a "mere" 141,125 digits smaller than the Mersenne prime found last August.
Would it not be reasonable to think that there are higher prime numbers, in fact wouldn't it be reasonable to think that if you keep on going there would be an infinite amount of prime numbers?

There are an infinite number of prime numbers. The hard (or slow, depending which approach you take) part is how to prove it's prime.

 

[CP5670]

As mentioned already, it's easy to show that there are infinitely many primes, but it's actually still not known if there are infinitely many Mersenne primes.

 

[KIAman]

As mentioned already, it's easy to show that there are infinitely many primes, but it's actually still not known if there are infinitely many Mersenne primes.

Actually, the hypothesis is that there are infinite amount of Mersenne primes considering infinite numbers but it becomes hard to verify IF a number is Mersenne prime or not beyond a certain size simply because of our lack of computing power.

 

[CP5670]

Actually, the hypothesis is that there are infinite amount of Mersenne primes considering infinite numbers but it becomes hard to verify IF a number is Mersenne prime or not beyond a certain size simply because of our lack of computing power.

That was my point, it's only a hypothesis so far.

The computational difficulty of finding out if a given number is a Mersenne prime or not is a different matter altogether.

 

[soccerballtux]

That was my point, it's only a hypothesis so far.

The computational difficulty of finding out if a given number is a Mersenne prime or not is a different matter altogether.

All the difficulty is in determining if it's a prime.
It's trivial to determine if the prime is Mersenne or not.

 

[CP5670]

All the difficulty is in determining if it's a prime.
It's trivial to determine if the prime is Mersenne or not.

The algorithms used to detect Mersenne primes actually take advantage of the special structure of those primes, and are more efficient than just using general primality tests.

In any case though, these algorithms don't tell us anything about whether there are infinitely many Mersenne primes or not.

 

[soccerballtux]

The algorithms used to detect Mersenne primes actually take advantage of the special structure of those primes, and are more efficient than just using general primality tests.

In any case though, these algorithms don't tell us anything about whether there are infinitely many Mersenne primes or not.

Ah, I see, that is what you were referring to.

 

[Cogman]

beyond the proof that there is an infinite number of primes, there is no other proof, AFAIK, about the existence of and infinite x type of primes. (twin primes, sexy primes, Mersenne, ect)

And there is a good reason for this. We don't have a formula to generate prime number x. It is very hard to determine what pattern prime numbers occur in. For all intents and purposes, it is pretty much random. (Though, we actually know the distribution of primes, oddly enough, it follows a logarithmic distribution)

 

[CP5670]

There is actually an exact formula for prime numbers in terms of the zeta function's zeros (the log estimate comes from it), but it's very complicated and useless for any numerical calculations. As you said, the small-scale behavior of the distribution of primes is seemingly quite random.