Will GIMPS Ever Discover a New Prime Through Doublecheck?

[jinydu]

The question says it all. What do you think?

[JuanTutors]

Isn't it true that 1% of first time LL tests end up being wrong?

[Uncwilly]

Originally Posted by dominicanpapi82

Isn't it true that 1% of first time LL tests end up being wrong?

I hope not, then there would be about 1000 more primes out there. About 1% have an incorrect risidual.

[smh]

Originally Posted by Uncwilly

I hope not, then there would be about 1000 more primes out there. About 1% have an incorrect risidual.

I haven't seen any analisys lately, but i seem to remember that this percentage is actually higher, and only gets higher as exponents get larger.

I remember something like 3-4%

[Mini-Geek]

Originally Posted by Uncwilly

I hope not, then there would be about 1000 more primes out there. About 1% have an incorrect risidual.

You're both right. 1% of them are wrong, meaning that they have an incorrect residual. Uncwilly just thought dominicanpapi82 meant that 1% are incorrectly saying that it's composite.

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.

[JuanTutors]

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.

[Mystwalker]

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.

[cheesehead]

Originally Posted by dominicanpapi82

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

What is meant by a set of exponents that "would" yield a Mersenne prime? That the set actually does include a Mersenne prime, or that we haven't yet determined for sure that it does not contain a Mersenne prime? (I'll assume the latter for my next remarks.)

actually give us at least one incorrect result goes above 50% only after about 69 exponents.

... but only if you remember that "incorrect result" does not mean that a number has been erroneously reported to be composite when it is actually prime.

As explained above, the 1% error rate refers to incorrect residues, not to a prime-versus-composite 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 LL-tested. 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.)

[JuanTutors]

Originally Posted by cheesehead

What is meant by a set of exponents that "would" yield a Mersenne prime?

What I meant was looking at the set of primes p such that Mp is actually a Mersenne prime.

[cheesehead]

Originally Posted by dominicanpapi82

What I meant was looking at the set of primes p such that Mp is actually a Mersenne prime.

Oh, so you mean that all the exponents in the set correspond to Mersenne primes.

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 L-L 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 LL-test 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%.

[retina]

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 (non-zero 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.