How do we know there is another Mersenne prime?

[kwstone]

I have only high school and engineering maths knowledge, so please correct me if I'm being really stupid here, but how do we know for sure that there are more Mersenne primes to be found?

I seem to recall reading that it is unknown whether there are infinitely many Mersenne primes. That means to me that it's quite possible that the number of them is finite. Could that finite number be 39? Or has it been (can it be?) proven that there must be at least a certain minimum number of Mersennes primes? Or are there any powerful arguments for the probability that they are infinite in number, even if we can't prove it right now?

I'd just like to be reassured that we are not all looking for a needle in a haystack that isn't there.

I belive it can be disproven that they are finite.
You might want to see FAQ here, and also see Chris Caldwell's Prime pages. Although you bring up a fundamental point, that if the speed of computers does not keep up, with the expected density, then the haystack will become to big to search efficiently. Because the gaps between get very large.

[wblipp]

Originally Posted by kwstone

Or are there any powerful arguments for the probability that they are infinite in number, even if we can't prove it right now?

From the prime number theorem, we know the probablity that numbers near the value "x" are prime is about 1/ln(x). So the probability that numbers near (2^p-1) are prime is about (1/(p*ln(2)). If we add this up for all primes, the expected number of Mersenne primes would be about (1/ln(2)) * [SUM(1/p)]. Some adjustments are appropriate because of known relationships such as M(p) is never divisible by 3, but this still leaves an estimate of the form c*[SUM(1/p)]. We know the sum increases without bound, so we expect there to be an infinite number of Mersenne primes.

[cheesehead]

Originally Posted by kwstone

I have only high school and engineering maths knowledge, so please correct me if I'm being really stupid here, but how do we know for sure that there are more Mersenne primes to be found?

We don't know, for sure.

I seem to recall reading that it is unknown whether there are infinitely many Mersenne primes.

That is correct - it is not yet known.

Some folks have conjectured that there are an infinite number. There are various arguments that it seems "reasonable" that there are an infinite number of Mersenne primes. But no one has ever presented a proof one way or the other.

That means to me that it's quite possible that the number of them is finite. Could that finite number be 39? Or has it been (can it be?) proven that there must be at least a certain minimum number of Mersennes primes?

No one has presented any proof that there are a certain minimum number (other than, obviously, that there are at least as many as have already been discovered).

Occasionally someone will humorously (or humourously, depending on which version of English he learned) predict that there are a finite number and that we have already found them all. For example, in 1998 just after M37 was discovered, one person posted on the Mersenne mailing list an assertion that there are only 37 Mersenne primes and that no more would ever be discovered.

I'd just like to be reassured that we are not all looking for a needle in a haystack that isn't there.

There's no guarantee -- it's an adventure.

But our success rate is lots higher than Seti@home's, so far.

[cheesehead]

Originally Posted by TTn

I belive it can be disproven that they are finite.

Please let us know when and where you substantiate your belief by finding such a disproof. :)

Although you bring up a fundamental point, that if the speed of computers does not keep up, with the expected density, then the haystack will become to big to search efficiently. Because the gaps between get very large.

On the other hand, if the number of Mersenne primes in existence above M38 meets or even exceeds our current expectation of density, or if someone develops a workable way to narrow the search, then GIMPS will keep processing that haystack just fine. :)

Please let us know when and where you substantiate your belief by finding such a disproof

I thought there was a section at Chris Caldwell's site.
I know that Guy's law says, not all Mp are square-free eventually.

On the other hand, if the number of Mersenne primes in existence above M38 meets or even exceeds our current expectation of density, or if someone develops a workable way to narrow the search, then GIMPS will keep processing that haystack just fine.

True, if so but then we take this to infinity, where there will be larger and larger gaps in GIMPS usefull activity due to operational challenges, and there will also be larger periods of very usefull activity, as in recent years, due to operational overcoming.

[eepiccolo]

Originally Posted by TTn

True, if so but then we take this to infinity, where there will be larger and larger gaps in GIMPS usefull activity due to operational challenges, and there will also be larger periods of very usefull activity, as in recent years, due to operational overcoming.

But we can't take it to infinity, because the world has to expire some day. But I guess then we'll be crunching in our space colonies :D :( 8)

Starting with perfect numbers

6 = 1+2+3
28 = 1+2+4+7+14
496 =1 +2+4+8+16+31+62+124+248

Since any any constellation of primes is possible then there will always be a mersenne prime q= -1(mod 2^(p-1))

[cheesehead]

Originally Posted by TTn

I thought there was a section at Chris Caldwell's site.

I've never seen one that said there was any proof that the number of Mersenne primes was infinite.

I know that Guy's law says, not all Mp are square-free eventually.

Oh? Can you link to that, or quote it entirely? And show how that disproves that the number of Mersenne primes is finite?

True, if so but then we take this to infinity, where there will be larger and larger gaps in GIMPS usefull activity due to operational challenges, and there will also be larger periods of very usefull activity, as in recent years, due to operational overcoming.

Or, simplifying, you got in another "dig" at GIMPS. :)

Starting with perfect numbers

6 = 1+2+3
28 = 1+2+4+7+14
496 =1 +2+4+8+16+31+62+124+248

Since any any constellation of primes is possible then there will always be a mersenne prime q= -1(mod 2^(p-1))

Will you please state your assertion more clearly?

Cheeshead,
There is only conjecture, of infinite primes, I did think it was proof.

As devil's advocate, Guy(1994) wolfram, is a mention of non square-free Mp. Which could point to finite, but vaguely.

Or, simplifying, you got in another "dig" at GIMPS.

No, actually quite the opposite, the original question is posed at M39 with one haystack. Of which has been recently loaded with needles due to George Woltman!

It is a good question if mankind will continue at all, with such misunderstandings. :(

[David John Hill Jr]

Anyone wish to challenge the results of 'continuing sequence, below by Hill'.
It has to always include every 2^x power. From it the trick may be to
test the recurring ,leaving out some 2^x per Mersenne included, to look for interspersed mersennes.
I call it the redundant case,personally.But there is no upper limit.