PRP on OBD candidates

[kriesel]

Currently, it would be premature to attempt a PRP on a gigadigit Mersenne number with prime exponent.

Only one has been taken in TF to 91 bits, none to 92 bits, and none have been P-1 factored to adequate bounds.

Existing primality testing software either does not support such large exponents, does not support PRP, or does not support PRP proof generation. The primality testing effort per candidate is so huge that IMO a first attempt should not be attempted until proof generation is available at such large exponents and fft lengths, and has been well tested for the same software and version at lower exponents.

An additional obstacle is achievable run time. To complete a PRP of an OBD in a year, the iteration benchmark would need to be under ~9.5 msec/iteration. 1 year / OBD PRP ~ 365 * 24 * 3600 seconds / (10⁹/log₁₀2) iterations = 0.009,49 seconds / iteration.

Versions of Mlucas available that can nominally handle PRP testing gigadigit candidates lack PRP proof generation.
While it is likely timings on native Linux would be better, on Ubuntu atop WSL, Mlucas V20.x produces benchmark timings for the required 192M fft length of:
Xeon Phi 7250 msec/iter = 1179.13
Xeon Phi 7210 msec/iter = 1156.88
i7-1165g7 msec/iter = 1153.63
dual-Xeon-e5-2690 msec/iter = 785.14
dual-Xeon-e5-2697v2 msec/iter = 404.89
The fastest of the above CPU msec/iter values corresponds to a run time of ~42.65 years.
It's likely that they would do better on native Linux. But not likely two orders of magnitude better.

Versions of gpuowl available that can nominally handle PRP testing gigadigit candidates lack PRP proof generation.
(V6.5-84 to 6.11-224) Run time on a Radeon VII would be around 50. msec/iter x 3.32E9 iterations ~ 5.3 years.

Verification by double-check would take equally long as a first test, in the absence of PRP proof generation capability or certification software capable of handling gigadigit Mersennes.

PRP proof generation space required will be considerable. Temporary space doubles for each increment of proof power and is proportional to exponent. Optimal proof power is an increasing function of exponent. Proof power 12 OBD temporary files would require ~ 1696. GiB each, judging by prime95 documentation. The resulting proof file would require ~5.4 GiB. Interim residue files are ~405. MiB each, as in Mlucas P-1.

[kriesel]

Still true:

Originally Posted by kriesel

Currently, it would be premature to attempt a PRP on a gigadigit Mersenne number with prime exponent.

Only three have been taken in TF to the sufficient level 92 bits, and while all three have been started in P-1 factoring, none have finished P-1 factoring yet to adequate stage 1 bounds, or started stage 2.
https://www.mersenne.ca/obd
The rest of post one is still current as of today.

[rainchill]

I'd love to test an OBD before I'm gone.

In theory, the LL double check would be faster as over the years it takes to run the test, newer hardware would be available to run the double check. Heck, as faster hardware becomes available it can be upgraded to during the test to speed along testing.

[kriesel]

Originally Posted by rainchill

I'd love to test an OBD before I'm gone.

In theory, the LL double check would be faster

PRP and LL test time are indistinguishable on same exponent and hardware (ratio <~1.001). However, the GEC only applies to PRP, not LL. Run times are so long on current best hardware and software that an OBD LL is almost certain to be wrong, unless run double with frequent interim residue comparison along the way. PRP with proof is not yet available for such large exponents. So primality testing OBD remains for now a fool's errand. So for now we prepare candidates, and wait for software to extend to cover this regime with PRP/GEC/proof. Advocating LL for primality testing indicates a lack of understanding of the error checking state of the art and requirements for such a large task. It's a bad choice at 100Mdigit or even 100M exponent, and worse, the higher the exponent. See https://www.mersenneforum.org/showpo...6&postcount=14
A leading candidate for OBD PRP with proof generation is a future release of Mlucas. Gpuowl extension is another possibility.

[slandrum]

Originally Posted by rainchill

I'd love to test an OBD before I'm gone.

In theory, the LL double check would be faster as over the years it takes to run the test, newer hardware would be available to run the double check. Heck, as faster hardware becomes available it can be upgraded to during the test to speed along testing.

Within a decade there will be faster hardware and better software much more suitable for testing the OBD candidates. Attempting to start a full test now would be a total waste of time and effort. Wait a few years until things catch up a bit.