20030629, 15:18  #1 
Jun 2003
Shanghai, China
155_{8} Posts 
How do we know there is another Mersenne prime?
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. 
20030629, 17:16  #2 
13·17·29 Posts 
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. 
20030629, 17:44  #3  
"William"
May 2003
New Haven
2,371 Posts 
Re: How do we know there is another Mersenne prime?
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20030629, 17:45  #4  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
Re: How do we know there is another Mersenne prime?
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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. Quote:
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. Quote:
But our success rate is lots higher than Seti@home's, so far. 

20030629, 18:08  #5  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}·3·641 Posts 
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20030702, 19:14  #6  
2^{13} Posts 
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I know that Guy's law says, not all Mp are squarefree eventually. Quote:


20030702, 19:32  #7  
Dec 2002
Frederick County, MD
101110010_{2} Posts 
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20030702, 19:34  #8 
2437_{16} Posts 
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^(p1)) 
20030702, 23:51  #9  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}·3·641 Posts 
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20030703, 02:32  #10  
7·757 Posts 
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 squarefree Mp. Which could point to finite, but vaguely. Quote:
It is a good question if mankind will continue at all, with such misunderstandings. :( 

20030713, 04:58  #11 
Jun 2003
Pa.,U.S.A.
2^{2}·7^{2} Posts 
A redundant case, below
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. 
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