
View Poll Results: Will GIMPS Ever Discover a New Prime Through Doublecheck?  
Yes  29  45.31%  
No  35  54.69%  
Voters: 64. You may not vote on this poll 

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20061002, 03:14  #1 
Dec 2003
Hopefully Near M48
2×3×293 Posts 
Will GIMPS Ever Discover a New Prime Through Doublecheck?
The question says it all. What do you think?

20061002, 12:44  #2 
Mar 2004
2^{2}·11^{2} Posts 
Isn't it true that 1% of first time LL tests end up being wrong?

20061002, 13:17  #3 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2·3^{3}·149 Posts 

20061002, 15:43  #4  
"Sander"
Oct 2002
52.345322,5.52471
29×41 Posts 
Quote:
I remember something like 34% 

20061002, 20:00  #5  
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
1000010101011_{2} Posts 
Quote:
I voted that another prime wouldn't be found, btw, since saying there is another prime is saying that out of the already extremely low chance of any exponent being prime, you add that a computer already, with 99% certainty, found that it's composite. 

20061003, 02:38  #6 
Mar 2004
2^{2}·11^{2} Posts 
Well if the probability of an erroneous result stays at 1%, then the probability that a given set of exponents that would yeild a Mersenne primes actually give us at least one incorrect result goes above 50% only after about 69 exponents.

20061003, 10:20  #7 
Jul 2004
Potsdam, Germany
3×277 Posts 
So far, only 44 such exponents have been found. I'd argue that only the last few (say, those 10 found by GIMPS) have the 1% error rate; maybe it was lower in the beginning.
Hence, we would have a way less than 50% chance that we've already missed a prime. 
20061008, 06:24  #8  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
Quote:
Quote:
As explained above, the 1% error rate refers to incorrect residues, not to a primeversuscomposite determination. Such errors will almost always (99.999+ %) be a matter of two different nonzero residues (composite either way), not a nonzero residue (composite) vs. a zero residue (prime). If there's no evidence that an incorrect nonzero residue is more or less likely than average to correspond to a zero residue when corrected, then it's just as if that number had not been LLtested. So the probability of finding a Mersenne prime among exponents whose only LL tests were incorrect is just the same as the probability of finding a Mersenne prime among numbers of similar size that have not been tested at all. That implies that about 1% of all Mersenne primes (assuming the error rate stays 1%) may be found among exponents whose first LL tests were reported with an incorrect residue. (However, I stll voted "No" in the poll above because I think there's a high probability that GIMPS will dissolve before a Mersenne prime is found by a DC.) Last fiddled with by cheesehead on 20061008 at 06:40 

20061009, 06:31  #9 
Mar 2004
111100100_{2} Posts 

20061010, 00:34  #10  
"Richard B. Woods"
Aug 2002
Wisconsin USA
7692_{10} Posts 
Quote:
Yes, in that case, as you wrote, "the probability that a given set of exponents that would yield a Mersenne primes actually give us at least one incorrect result goes above 50% only after about 69 exponents". But that's true of any set of exponents whether or not any exponent in the set corresponds to a Mersenne prime, as long as "incorrect result" is understood to mean "incorrect residue". That is, for any set of 69 or more LL tests each of which has an error probability of 1%, the probability of at least one erroneous result (residue) from that set is above 50%. So, getting back to your special case of all exponents corresponding to Mersenne primes, if we LLtest all Mersenne primes from M45 through M113 once each, the chance that one of those residues will be erroneous (= nonzero, in this special case), and thus the chance that the primality would be discovered only on a doublecheck (or later), is greater than 50%. Last fiddled with by cheesehead on 20061010 at 01:28 

20061010, 14:50  #11 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2^{3}×3×227 Posts 
So far 10 primes found, so only 59 primes to go to get >50% chance of a bad result returned for a real prime. So after M103rd we expect >50% chance that one was missed by the first time check? All other exponents can be ignored because false positives (ie. zero residual) are excluded and incorrect negatives (nonzero but wrong residual) are discarded.
To answer the question given in the topic title, at a rate of 1 prime discovered each year (the current average rate) that would be 59 years to give >50% probability. So I predict year 2065 will be the year a DC finds a prime. 
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