DC chance to find Mersenne Prime

[houding]

Just out of curiosity.

Not taking into account different RES values between first and DC run, what is the chance that a DC run might actually show an exponent is a Mersenne Prime whereas the first run did not? Or will a prime always be found in the first run and only proofed in the DC run?

[Mini-Geek]

Originally Posted by houding

Just out of curiosity.

Not taking into account different RES values between first and DC run, what is the chance that a DC run might actually show an exponent is a Mersenne Prime whereas the first run did not?

If the RES values aren't different, and the first showed it was composite? 0, that is, no chance at all...by definition. The whole point of the DC is that if the RES value in the first run was wrong, there's a chance that the number is prime. If the residue of the first run is correct, then the number is composite. If the residue of the first run is incorrect, then the number just might be prime has just a good a chance of being prime as if it had no test had been run.

Originally Posted by houding

Or will a prime always be found in the first run and only proofed in the DC run?

No, DCs have a very real chance of finding a prime number. It's a small chance, but it exists. Most often, the prime is discovered in the first run and verified (outside of the normal DC process) quickly. DCs (as you probably think of them) are for numbers that came up composite in the first test. In order to verify that result, we need a matching residue from a DC.

If you multiply the probability of the first test being wrong (usually ~4%, or 0.04) by the probability of the number being a Mersenne prime based only on the factoring work done (very small), you get the probability that a DC will result in a Mersenne prime.

This can work out to something like 1 in 14584730 (which is the probability for a DC I just reserved to check it). Prime95 tells you this for numbers it's testing in its Test > Status window.