Developer's corner

[kriesel]

For purposes of this post, "large exponents" means well ahead of the first test wavefront, which is at ~102M-114M as of March 2021.
Note, while both PRP and LL interim residues are listed here, PRP with GEC and proof generation should be used for large exponents and first-test wavefront exponents whenever practical. High proof powers, at least 8 are recommended. Proof generation enables verification at ~1% or less the effort of a traditional full double check. GEC essentially eliminates error in final residue results. At 100Mdigit, a small sample of results of LL & DC & TC indicates an error rate of about 20% per test. That is consistent with error proportional to run time. Extrapolating to higher exponent indicates high probability of error near 1G. (Run time of 999M is (999/332)^2.1= 10.1 times longer; 1 - (1-0.20)^10.1 = 0.89, 89% probability of bad result.)


~50Mdigit interim residues
166000013 from a complete PRP3 with proof certified valid (49,970,984 decimal digits) FFT: 9M 1K:9:512 (17.59 bpw)
gpuowl-win-v6.11-364-g36f4e2a
2020-08-09 04:21:37 asr2/radeonvii0 166000013 OK 800 0.00%; 1418 us/it; ETA 2d 17:23; 9c30f47224fb8783 (check 0.96s)
2020-08-09 04:26:20 asr2/radeonvii0 166000013 OK 200000 0.12%; 1417 us/it; ETA 2d 17:15; c6ae0c584288f318 (check 0.94s)
2020-08-09 04:45:17 asr2/radeonvii0 166000013 OK 1000000 0.60%; 1417 us/it; ETA 2d 16:56; 8d41ce621b768b00 (check 1.03s)
2020-08-09 08:18:39 asr2/radeonvii0 166000013 OK 10000000 6.02%; 1416 us/it; ETA 2d 13:22; b8fba9175797f5ed (check 0.95s)
2020-08-11 07:15:04 asr2/radeonvii0 166000013 OK 100000000 60.24%; 1418 us/it; ETA 1d 02:00; b00fa62adb2d712c (check 0.95s)


100Mdigit interim residues
mlucas v17.0 LL 1 core of a xeon e5645 (Not recommended; at these timings, actually msec not sec/iteration, a full run is over 8 years; v17.0 did not include the Jacobi check. The odds of correct completion over such a long run are negligible.)
332220523 100,008,343 decimal digits
[Jul 22 13:31:39] M332 220 523 Iter# = 10000 [ 0.00% complete] clocks = 02:16:04.515 [816.4516 sec/iter] Res64: 1A313D709BFA6663. AvgMaxErr = 0.171224865. MaxErr = 0.250000000.
[Jul 23 09:51:21] M332220523 Iter# = 100000 [ 0.03% complete] clocks = 02:16:05.131 [816.5132 sec/iter] Res64: 91B688264B5B3F39. AvgMaxErr = 0.171926060. MaxErr = 0.250000000.
[Jul 31 21:47:38] M332220523 Iter# = 1000000 [ 0.30% complete] clocks = 02:15:44.087 [814.4088 sec/iter] Res64: 7CC62737AA46CDF8. AvgMaxErr = 0.171821204. MaxErr = 0.234375000.
[Oct 26 19:22:44] M332220523 Iter# = 10000000 [ 3.01% complete] clocks = 02:20:19.131 [841.9132 sec/iter] Res64: 2B8CC43403D28DAC. AvgMaxErr = 0.171851920. MaxErr = 0.250000000.

DoubleCheck=332220523
gpuowl v6.11-292 (manageable projected total time at 15 days on a Radeon VII)
2020-05-24 20:41:20 asr2/radeonvii3-w2 332220523 LL 100000 0.03%; 3843 us/it; ETA 14d 18:33; 91b688264b5b3f39
2020-05-24 21:39:36 asr2/radeonvii3-w2 332220523 LL 1000000 0.30%; 3870 us/it; ETA 14d 20:03; 7cc62737aa46cdf8
(version?)
2020-07-30 09:48:45 asr2/radeonvii2 332220523 LL 2000000 0.60%; 3722 us/it; ETA 14d 05:25; 627dfa42f70bd3b4
(LL residues above for M332220523 are otherwise unverified; passed Jacobi checks where applicable)

gpuowl v6.0-b7bb1c3 PRP 332220523 on an RX480
2019-02-04 23:30:02 condorella/rx-480 332220523 OK 10000 0.00%; 16.52 ms/sq; ETA 63d 12:29; 503cd91d7b8e30e5 (check 7.48s)
continuation on gpuowl v6.10-9-g54cba1d
2021-03-15 18:04:59 332220523 100000 0.03%; 18714 us/sq; ETA 71d 22:30; 951c94f813216db9
continuation on gpuowl v6.11-380-g79ea0cc
2021-03-15 18:33:02 condorella/rx480 332220523 OK 200000 0.06%; 14796 us/it; ETA 56d 20:39; 6cd7d19bb77ad049 (check 6.58s)
2021-03-15 19:22:27 condorella/rx480 332220523 OK 400000 0.12%; 14793 us/it; ETA 56d 19:30; b12cf8adffda122c (check 6.56s)

gpuowl (version?) independent run on a 5700xt:
2020-09-03 13:13:03 asr2/5700xt 332220523 OK 200000 0.06%; 7520 us/it; ETA 28d 21:34; 6cd7d19bb77ad049 (check 4.05s)


~gigabit or 300Mdigit interim residues (mostly unverified, except for green)
CUDALucas v2.06 on gtx1080ti (LL, no Jacobi check, unverified; not recommended at 1.4 years estimated duration on a GTX1080Ti)
999999937 301,029,977 decimal digits
| Jan 09 04:48:51 | M999999937 10000 0x567ad47461d3bb5f | 57344K 0.21875 41.1682 41.16s | 473:21:41:27 0.00% |
| Jan 20 08:14:38 | M999999937 100000 0xe776f4a0dcd3491d | 57344K 0.18750 44.3713 44.37s | 506:19:41:57 0.01% |
| Jan 20 19:26:39 | M999999937 1000000 0x141a108c13a86d5a | 57344K 0.18750 45.3946 45.39s | 516:19:59:55 0.10% |
| Jan 23 07:03:54 | M999999937 5000000 0x0811f10855dab84c | 57344K 0.20313 44.9825 44.98s | 516:10:56:38 0.50% |


gpuowl v6.11-380-g79ea0cc independent run from start on gtx 1080
2021-03-27 14:28:07 gpuowl v6.11-380-g79ea0cc
2021-03-27 14:29:16 asr3/gtx1080 999999937 LL 1000 0.00%; 52191 us/it; ETA 604d 01:29; ddadfed64e080856
(Jacobi check on exit passed; continuing:)
2021-03-27 14:55:18 asr3/gtx1080 999999937 LL 10000 0.00%; 50442 us/it; ETA 583d 19:32; 567ad47461d3bb5f
2021-03-27 15:03:46 asr3/gtx1080 999999937 LL 20000 0.00%; 50782 us/it; ETA 587d 17:50; 78a2f270a1bba92d
2021-03-27 15:12:14 asr3/gtx1080 999999937 LL 30000 0.00%; 50794 us/it; ETA 587d 21:05; d79b942904525426
2021-03-27 15:20:42 asr3/gtx1080 999999937 LL 40000 0.00%; 50823 us/it; ETA 588d 05:02; 650ca8c106ac7d12
2021-03-27 15:29:11 asr3/gtx1080 999999937 LL 50000 0.01%; 50869 us/it; ETA 588d 17:34; 98420630c9a4b877
2021-03-27 15:37:39 asr3/gtx1080 999999937 LL 60000 0.01%; 50850 us/it; ETA 588d 12:03; adb31fa7cf23b0ba
2021-03-27 15:46:08 asr3/gtx1080 999999937 LL 70000 0.01%; 50847 us/it; ETA 588d 11:13; 20b88580b0d5c8a5
2021-03-27 15:54:38 asr3/gtx1080 999999937 LL 80000 0.01%; 51002 us/it; ETA 590d 06:05; 2b4ab3e44fcbdc84
2021-03-27 16:03:07 asr3/gtx1080 999999937 LL 90000 0.01%; 50957 us/it; ETA 589d 17:20; b9c6eeca4e553904
2021-03-27 16:11:37 asr3/gtx1080 999999937 LL 100000 0.01%; 50972 us/it; ETA 589d 21:24; e776f4a0dcd3491d

CUDALucas V2.06beta on V100 (LL, no Jacobi check, unverified, ~6 months; interim residue extracted by ramgeis from an interim save file using a perl program provided by Kriesel)
999999937 Iteration 110,000,000 Residue64 0x87ded9d3ab3e7f38
See also https://mersenneforum.org/showthread...859#post495859


The PRP equivalent is feasible marginally with Gpuowl on a Radeon VII, at ~0.51 years to complete; LL in Gpuowl on a Radeon VII is also possible but not recommended since it's only protected by an occasional Jacobi check's ~50% chance of detection of an error. Such long runs are likely to go wrong with undetected errors.
Gpuowl v6.2-e2ffe65 PRP on RX480:
2019-02-08 17:48:10 condorella/rx480 999999937 10000 0.00%; 72.02 ms/sq; ETA 833d 14:01; a30a0c45e9fb828c
2019-02-08 21:16:26 condorella/rx480 999999937 100000 0.01%; 74.38 ms/sq; ETA 860d 18:51; 3efc806b68d92b86
continuation on gpuowl v6.11-380-g79ea0cc on RX480, quite a speed improvement, 1.51X over v6.2, but still not recommended at ~1.56 years on an RX480:
2021-03-16 12:20:54 condorella/rx480 999999937 OK 1000000 0.10%; 49236 us/it; ETA 569d 07:00; e1b53ccf581f928b (check 22.08s)


1.5Gbit interim residues
gpuowl v4.6 PRP on RX480 (not recommended, at estimated 7.8 years to completion; also moot since it has a known small factor) Following is combined PRP/P-1, type 0, B1 bounds 0, so presumably base = 3.
2018-11-01 16:35:37 condorella-rx480 10000/1500000041 [ 0.00%], 164.51 ms/it [163.50, 174.58]; ETA 2856d 01:42; 4fc4ebb3728095f7
CUDALucas 2.06 fails on 1,500,000,043 with an error similar to that for 2Gbit (see below).


2Gbit interim residues
P-1 stage 1 appears possible on gpuowl v6.11-380, at over 6 days on a Radeon VII:
2021-03-26 13:57:40 asr2/radeonvii4 2147483563 P1 B1=11000000, B2=600000000; 15869712 bits; starting at 0
2021-03-26 14:02:48 asr2/radeonvii4 saved
2021-03-26 14:03:21 asr2/radeonvii4 2147483563 P1 10000 0.06%; 34113 us/it; ETA 6d 06:17; d78cbf554970d8c1

https://www.mersenne.ca/exponent/2147483743 Projected PRP time on a Radeon VII is 2.63 years, not recommended. Gpuowl V6.11-380 PRP interim residues with timings:
2021-03-26 12:43:44 asr2/radeonvii4 2147483743 OK 20000 0.00%; 38636 us/it; ETA 960d 07:01; 304a08968e896b20 (check 23.14s)
2021-03-26 13:36:46 asr2/radeonvii4 2147483743 OK 100000 0.00%; 38636 us/it; ETA 960d 05:56; 0637bf5d33611a9d (check 22.85s)


gigadigit ~3.3gigabit interim residues
Gpuowl v6.5-84 PRP3 (not recommended at an estimated 6.3 years on a Radeon VII; lacks PRP proof capability)
M3,321,928,097 1,000,000,001 decimal digits (This one is moot as it has multiple known small factors)
FFT 196608K: Width 512x8, Height 256x8, Middle 12; 16.50 bits/word
2020-05-24 17:21:43 radeonvii3 3,321,928,097 20000 0.00%; 60053 us/sq; ETA 2308d 21:54; 1a05f0ca51fb8e7a
2020-05-24 18:41:56 radeonvii3 3,321,928,097 100000 0.00%; 60122 us/sq; ETA 2311d 12:24; 5b2cb77f57840bcc


Top of this thread https://www.mersenneforum.org/showthread.php?t=24003
Top of reference tree: https://www.mersenneforum.org/showpo...22&postcount=1

[kriesel]

See also Ernst's post on shift in Mlucas.

Think of the primality testing software's math routines as an emulation of a very wide word calculator. It dynamically adjusts to p bits width, where p is the prime exponent of Mp=2^p-1.

1) All the LL tests should use the same (before shifted) seed value, S₀=4, not ten or the other known seed values, or bad seed values such as assorted other powers of two, so that the composite numbers' final test residues will be comparable. That's separate from considerations of shift.
See https://www.mersenneforum.org/showpo...12&postcount=6

2) A very simple example with zero shift:
p=7, M7=127
s₀=4 = 100.b
shift = 0 so the value in our emulated wide-word calculator is 4.
S₁= 4^2-2 = 14 = 1110.b
shift does not change from one iteration to the next.
S₂=14^2-2 =194 = 1 1000010 mod M7 = 67 = 1000011.b

3) Second run, same simple example except with shift 1:
p=7
s₀=4
startingshift = 1 so the value in our simulated fft calculator lands one bit left, looking like 8 = 1000 but representing 100.0b. At this point previousshift=startingshift.
square it, compute the new shift, and apply a shifted -2:
8^2 =64 = 1000000 but representing 10000.00b.
shift=previousshift * 2 = 2
subtract 2 <<shift = 8, leaving 56 =111000 (representing 14 shifted 2 left, 1110.00b)
Mod Mp is relative to that shifted "."
for the next iteration, shift = previousshift *2=4, and repeat.
S₂=56^2 - 2<<4 = 3104 mod M7<<4 = 1072 = 1000011.0000b (67<<4)
But wait, it's actually wrapped in p=7 bits (Mod M7): 1000 011.0000b = 011. 1000 which is not 3.5, it's a wrapped representation of a 4<<shifted 67.

4) What the programs supporting nonzero shift will report in their test result as shift is startingshift. But they will retrieve the residue from the simulated calculator beginning at lsb = the final shift computed after the last iteration, doing the final de-shift and de-wrap then. It's more complicated than just doubling shift as in the preceding, because the unit point wraps around, many times.

5) In practice, for programs that implement shift (offset), the initial shift is chosen as pseudorandom in the range 1 to p-1. (I think that was 0 to p-1 at one point, but got changed to reduce the small chance of separate runs coinciding at 0 shift. Shifting by p bits is equivalent to no shift, so should also be excluded. Some applications, and some versions of prime95, cCUDALucas, and Gpouwl, did not implement shift so were always 0 shift.) So about p/2 on average would be the initial shift.
For shift=3, 5 iterations for p=7 give successive iteration shifts of 6, 12, 24, 48, 96 mod 7 = 5 for recovery of the final residue.

6) Why go to all this trouble of shifting and its extra accounting? Because sometimes in using double precision floating point fft to rapidly emulate a monstrously-wide-word integer calculator, for p in the many millions, the roundoff error affects the outcome, or other errors occur, and having the data shifted differently relative to the 16 - 19 bits/word boundaries causes the errors to behave differently, and sufficiently so for their effect to change the final result. It can also cause different behavior from software bugs that are data-dependent. Including, ironically, a bug in the shift code with a one in a million chance of producing wrong residues. Or here.
The use of pseudorandom offset is one of a number of techniques to avoid, detect, or detour around numerical error. And result reporting error causing duplicate reporting from a single run, leading to a singly tested exponent being falsely recorded as double-checked with match, can be detected because of matched shift values.

7) The authors of the major GIMPS primality testing software could give a clearer explanation and probably have. Ernst's Odroid magazine article may include one. George, Mihai, and Ernst have posted about these matters in the prime95 and gpuowl and Mlucas development threads.

8) I think the list of "shifty" GIMPS programs includes prime95/mprime beginning by v17, CUDALucas beginning v2.05, Mlucas beginning v18, and rare early gpuowl versions, and the list of "shiftless" GIMPS programs includes cllucas, early CUDALucas, most versions of gpuowl (whether implementing LL or PRP or both), and Mlucas before V18. Shift is useful in primality testing which would get done more than once. It is not useful in computations that typically get done once, such as TF or P-1. Shift is known to be not present in prime95 P-1 factoring, CUDAPm1, and gpuowl P-1.


Top of this thread https://www.mersenneforum.org/showthread.php?t=24003
Top of reference tree: https://www.mersenneforum.org/showpo...22&postcount=1

[kriesel]

Server Support

TF
Up to 1G exponent at mersenne.org; 1G to 10G at mersenne.ca

P-1
Up to 1G exponent at mersenne.org

LL
Up to 1G exponent at mersenne.org

PRP
Up to 1G exponent at mersenne.org, although note, currently the server is limited by its SSE2 instruction set to PRP proof file processing up to ~595.7M automatically

Software availability

TF
Mfaktx are capable of up to 2³²-1 (largest prime exponent 4,294,967,291). Mfakto for AMD GPUs or Intel IGPs or AMD IGPs, Mfaktc for CUDA (NVIDIA) GPUs.

P-1
Mlucas current release does not support P-1. The capability is being added.

CUDAPm1 is theoretically capable of up to 2³¹-1 (largest prime exponent 2,147,483,647) but in practice may reach to ~100Mdigit on a reasonably modern NVIDIA GPU.

Mprime/prime95 are capable of up to varying exponents depending on CPU type, with maximum 64K fft length on AVX512 allowing ~1169M.

Various versions of Gpuowl support P-1 either standalone or as part of a PRP run (but not both). The fft lengths supported are one determinant of the maximum exponent. The maximum exponent for P-1 may be less than for PRP, depending at least upon bounds and available GPU or system memory, and a bit more requirement for carry handling reducing the available bits/word on a given fft. No gpuowl version to date supports gigadigit exponent P-1. I have performed complete P-1 runs to mersenne.ca recommended GPU72 bounds on local Radeon VII and Colab provided P100 GPUs up to 1Gbit. Current versions of gpuowl support fft lengths consistent with P-1 to about exponent 2³¹-1. Most P-1 implementations are without offset, Jacobi symbol check, or GEC. GpuOwl V7.x supports both some GEC and Jacobi symbol checking during P-1 computations combined with PRP.

PRP
ONLY PRP with GEC & Proof should be considered for new first primality tests.
Mprime/prime95 are capable of up to varying exponents depending on CPU type, with maximum 64K fft length on AVX512 allowing up to ~1169M.
Gpuowl in relatively current and efficient versions is capable of 120M fft lengths, nominally capable of 2172.36M per its help output.

Gpuowl around V6.5-84 was capable of PRP on gigadigit exponents and a bit more, 3339.40M indicated in the help output.

Mlucas is capable with fft lengths up to 256M of up to 2³²-1 (largest prime exponent 4,294,967,291). V20 will support larger, including 512M fft lengths for F33 and so presumably up to ~M8,589,934,583 (which has a known factor). PRP proof capability is also being added.

LL
LL should be considered as for primality verification or reliability research purposes only, not suitable for ordinary first tests! Even its use in LLDC is questionable.

Mprime/prime95 are capable of up to varying exponents depending on cpu type, with maximum 64M fft length on AVX512 allowing ~1169M.

CUDALucas is nominally capable of up to 256M fft (for recent CUDA versions), which would be sufficient for ~2³²-1. However, its exponent variable is signed 32 bit, 2³¹-1 (largest prime exponent 2,147,483,647), capping exponent for fft lengths 128M and above. In practice it is limited further, to ~1.43Gbits, determined empirically, as shown below.

Gpuowl in relatively current and efficient versions is capable of 120M fft lengths, though there may be issues in its upper reaches also. Nominally that is capable of 2172.36M.
Gpuowl around V6.5-84 was nominally capable of gigadigit exponents but did not include LL then.

Mlucas is capable with fft lengths up to 256M of up to 2³²-1 (largest prime exponent 4,294,967,291). V20 will support larger, including 512M fft lengths for F33 and so presumably up to ~M8,589,934,583 (which has a known factor). Mlucas does not yet implement the Jacobi symbol check for LL testing.

Memory requirements

There are several requirements to consider. Disk space, system ram, and sometimes GPU ram; also how much data may be transferred to and from the PrimeNet server in the case of PRP proof and certification.

TF: minor

P-1: large
Gpuowl typically requires at least 16 or 24 buffers available on the GPU for P-1 stage 2. This limits the maximum exponent according to GPU memory capacity. On 16GB GPUs, up to 1G exponents can be run in v6.11-380, and at least slightly higher. (1,000,000,007 ran successfully to completion of both stages on a Radeon VII.)

PRP:
GPU-Z indicates usage of 2.7 GB GPU ram in Gpuowl v6.11-318 for exponent ~2³¹.
Interim save files will be at least the binary packed size of the exponent, plus some words for housekeeping, such as file format, exponent, iteration, etc. Most applications save one or two residues in a save file. Mprime / prime95 for some cases will store up to 3. So a rough bound for prime95 save files is 3/8 p bytes each.
The number of save files saved is controlled by each program's settings.
For temporaries stored on disk for proof generation, their total size and timing depends on the application, exponent, and proof power.
Mprime / prime95 pre-allocates space sufficient for all the temporaries. Gpuowl adds individual temporaries files as the PRP computation progresses.

LL:
Storage requirements are similar to PRP without proof.

Run time scaling

Run time scaling for some specific hardware and software combinations can be found in the relevant software-specific reference threads.
TF: exponential in bit level, inversely proportional to exponent for a given bit level. The overall effect since recommended bit level increases with exponent is run time increases with exponent. Large exponents can require weeks of TF on a fast GPU to complete recommended levels.
P-1: to optimal bounds, about 1/30 of primality testing time.
Primality testing: scales as approx exponent^2.1.

Hardware lifetime

Run times of large exponents may exceed the probable hardware useful life, or the user's remaining life expectancy. A good backup regimen and transfer to replacement hardware is a workaround, if the user has enough patience and human life expectancy remaining. Very long undertakings may also need a succession plan.

Software & Hardware combination reliability

Overall experience
Madpoo has checked error rate of primality tests on occasion. Most of the data points are at much smaller exponent than the current wavefronts. Run time was probably substantial at the time they were performed. Observed error rate was about 4% per exponent, 2% per test. These results date from before introduction to GIMPS testing of the LL Jacobi symbol check or PRP or GEC. (See https://mersenneforum.org/showpost.p...1&postcount=67)

100M exponent experience
For the period October 1 2020 to March 26 2021, the observed rate of LL final residue error per test for exponents between 97M and 103M was ~0.94%. Presumably most of these were performed with mprime / prime95 with the Jacobi check, and some were with ECC system ram. (See first attachment.)
Extrapolating to approximately triple the exponent, ten times the run time, projects ~10% error rate at 100Mdigit; to ten times the exponent, 126 times the run time, projects a very high error rate for single-run gigabit exponents, ~30% chance of correct result, 70% chance of error in the result. Extrapolating further to gigadigit, 12.44 times the runtime of a gigabit test, near certainty of error, ~0.37 ppm chance of correct result.
Longer term average error rate was found to be higher, at ~2.5% for exponents 97M-103M. This is probably reflective both of runs without Jacobi symbol check and longer run times on slower hardware in previous years. (See second attachment.)
More detail on observed error rates can be found at https://www.mersenneforum.org/showpo...40&postcount=4 and its links.

Limited data on 100Mdigit
There are very few verified exponents from which to assess 100 Mdigit LL test error rate. The available data indicates about 21.% error rate. Many of these were done long ago, probably with long run times and without the benefit of any Jacobi check. (See third attachment.)

Higher
There's very little data for above the 100Mdigit region. Most over 350M were done as paired LL runs with regular interim residue comparison, or as PRP with GEC. Extrapolations to gigabit / 300M digit indicate paired runs with frequent interim res64 or other, new error checks or approaches will be required for LL testing. (See fourth attachment.)

CUDALucas is Lucas-Lehmer test capable only, not PRP. It lacks Jacobi symbol check. The GEC that is so useful in keeping PRP runs reliable is not applicable to the LL residue sequence, so not applicable to CUDALucas. It is also slower than recent Gpuowl (V6.11-3xx, v7.x-y) on the same hardware and assignment. Avoid CUDALucas, use Gpuowl whenever possible.


100Mdigit experiments
Obtaining matching interim residues of 100Mdigit exponents is usually straightforward on available software and adequate reliability hardware. Matching interim residues were produced with independent runs on the first tries, among Mlucas and Gpuowl for LL, and Gpuowl for PRP. (See https://www.mersenneforum.org/editpo...&postid=546384 for details)


~gigabit or 300Mdigit experiments
Obtaining matching interim residues of 300Mdigit exponents involved 4 attempts to get two matching. Details follow; results are summarized at the above link.

interim residues (mostly unverified, except for green)
CUDALucas v2.06 on gtx1080ti (LL, no Jacobi check, unverified; not recommended at 1.4 years estimated duration on a GTX1080Ti)
999999937 301,029,977 decimal digits

|  Jan 09  04:48:51  | M999999937     10000  0x567ad47461d3bb5f  | 57344K  0.21875  41.1682   41.16s  | 473:21:41:27   0.00%  |
|  Jan 20  08:14:38  | M999999937    100000  0xe776f4a0dcd3491d  | 57344K  0.18750  44.3713   44.37s  | 506:19:41:57   0.01%  |
|  Jan 20  19:26:39  | M999999937   1000000  0x141a108c13a86d5a  | 57344K  0.18750  45.3946   45.39s  | 516:19:59:55   0.10%  |
|  Jan 23  07:03:54  | M999999937   5000000  0x0811f10855dab84c  |  57344K  0.20313  44.9825   44.98s  | 516:10:56:38   0.50%  |

CUDALucas v2.06 on gtx1080 (LL, no Jacobi check, unverified; not recommended at 1.85 years estimated duration on a GTX1080

|   Date     Time    |   Test Num     Iter        Residue        |    FFT   Error     ms/It     Time  |       ETA      Done   |
|  Mar 27  13:24:34  | M999999937     10000  0xd0ba62caab74a325  | 57344K  0.20313  58.3992  224.72s  | 671:06:41:49   0.00%  |
|  Mar 27  13:34:21  | M999999937     20000  0x3b0de44a71284153  | 57344K  0.20313  58.7176  587.17s  | 675:10:19:38   0.00%  |
|  Mar 27  13:44:08  | M999999937     30000  0xee988cb77112ea03  | 57344K  0.20313  58.7149  587.14s  | 676:19:10:58   0.00%  |
|  Mar 27  13:53:55  | M999999937     40000  0x906263cb2b36ac4c  | 57344K  0.20313  58.7086  587.08s  | 677:11:05:19   0.00%  |
|  Mar 27  14:03:43  | M999999937     50000  0x454a88b55988ce1e  | 57344K  0.21094  58.7162  587.16s  | 677:20:59:32   0.00%  |

(deleted this bad run's set of interim residue files)
Any residues produced after a known-bad residue (red or purple here) will also be bad. The grayed lines are preserved since they show iteration speed variation.


Gpuowl (which has the Jacobi check for LL) on RX480, not recommended at 1.56 years or more depending on gpuowl version; GTX1080 similar; Radeon VII marginally feasible at ~6 months

2019-02-08 17:34:48 gpuowl v6.2-e2ffe65
2019-02-08 17:34:48 condorella/rx480 -user kriesel -cpu condorella/rx480 -device 0 -fft +0 
2019-02-08 17:34:48 condorella/rx480 999999937 FFT 73728K: Width 256x8, Height 256x8, Middle 9; 13.25 bits/word
2019-02-08 17:34:48 condorella/rx480 using long carry kernels
2019-02-08 17:34:54 condorella/rx480 OpenCL compilation in 5333 ms, with  "-DEXP=999999937u -DWIDTH=2048u -DSMALL_HEIGHT=2048u -DMIDDLE=9u  -I.  -cl-fast-relaxed-math -cl-std=CL2.0"
2019-02-08 17:35:06 condorella/rx480 999999937.owl not found, starting from the beginning.
2019-02-08 17:37:08 condorella/rx480 999999937 OK      800  0.00%; 71.64 ms/sq; ETA 829d 05:22; c3c8e02da339fdfa (check 31.90s)
2019-02-08 17:48:10 condorella/rx480 999999937       10000  0.00%; 72.02 ms/sq; ETA 833d 14:01; a30a0c45e9fb828c
2019-02-08 21:16:26 condorella/rx480 999999937      100000  0.01%; 74.38 ms/sq; ETA 860d 18:51; 3efc806b68d92b86

gpuowl v6.10-9-g54cba1d on RX480 continuation from 102000 to 495600
2021-03-15 23:16:33 999999937 FFT 57344K: Width 256x8, Height 256x8, Middle 7; 17.03 bits/word

gpuowl v6.11-380-g79ea0cc on RX480 continuation

2021-03-16 06:23:01 condorella/rx480 999999937 FFT: 56M 4K:14:512 (17.03 bpw)
2021-03-16 06:23:01 condorella/rx480 Expected maximum carry32: 833C0000
2021-03-16 06:51:54 condorella/rx480 999999937 OK   600000   0.06%; 49222 us/it; ETA 569d 08:39; c974a7f641226218 (check 21.93s)
2021-03-16 09:36:25 condorella/rx480 999999937 OK   800000   0.08%; 49242 us/it; ETA 569d 11:23; 321dd77853313c0c (check 22.04s)
2021-03-16 12:20:54 condorella/rx480 999999937 OK  1000000   0.10%; 49236 us/it; ETA 569d 07:00;  e1b53ccf581f928b (check 22.08s)

(deleted this bad run's set of interim residue files)


gpuowl v6.11-380-g79ea0cc independent run from start on gtx 1080

2021-03-27 14:28:07 gpuowl v6.11-380-g79ea0cc
2021-03-27 14:28:07 config: -device 1 -user kriesel -cpu asr3/gtx1080 -maxAlloc 6500 -proof 9 -cleanup -yield -use NO_ASM
2021-03-27 14:28:07 device 1, unique id ''
2021-03-27 14:28:07 asr3/gtx1080 999999937 FFT: 56M 4K:14:512 (17.03 bpw)
2021-03-27 14:28:07 asr3/gtx1080 Expected maximum carry32: 833C0000
2021-03-27 14:28:16 asr3/gtx1080 OpenCL args "-DEXP=999999937u  -DWIDTH=4096u -DSMALL_HEIGHT=512u -DMIDDLE=14u -DPM1=0 -DCARRY64=1  -DWEIGHT_STEP_MINUS_1=0xf.57fb440c6997p-4  -DIWEIGHT_STEP_MINUS_1=-0xf.aa3b4ca84faap-5 -DNO_ASM=1   -cl-unsafe-math-optimizations -cl-std=CL2.0 -cl-finite-math-only "
2021-03-27 14:28:22 asr3/gtx1080

2021-03-27 14:28:22 asr3/gtx1080 OpenCL compilation in 6.10 s
2021-03-27 14:28:24 asr3/gtx1080 999999937 LL        0 loaded: 0000000000000004
2021-03-27 14:29:14 asr3/gtx1080 Stopping, please wait..
2021-03-27 14:29:16 asr3/gtx1080 999999937 LL     1000   0.00%; 52191 us/it; ETA 604d 01:29; ddadfed64e080856
(Jacobi check on exit passed; continuing:)
2021-03-27 14:55:18 asr3/gtx1080 999999937 LL    10000   0.00%; 50442 us/it; ETA 583d 19:32; 567ad47461d3bb5f
2021-03-27 15:03:46 asr3/gtx1080 999999937 LL    20000   0.00%; 50782 us/it; ETA 587d 17:50; 78a2f270a1bba92d
2021-03-27 15:12:14 asr3/gtx1080 999999937 LL    30000   0.00%; 50794 us/it; ETA 587d 21:05; d79b942904525426
2021-03-27 15:20:42 asr3/gtx1080 999999937 LL    40000   0.00%; 50823 us/it; ETA 588d 05:02; 650ca8c106ac7d12
2021-03-27 15:29:11 asr3/gtx1080 999999937 LL    50000   0.01%; 50869 us/it; ETA 588d 17:34; 98420630c9a4b877
2021-03-27 15:37:39 asr3/gtx1080 999999937 LL    60000   0.01%; 50850 us/it; ETA 588d 12:03; adb31fa7cf23b0ba
2021-03-27 15:46:08 asr3/gtx1080 999999937 LL    70000   0.01%; 50847 us/it; ETA 588d 11:13; 20b88580b0d5c8a5
2021-03-27 15:54:38 asr3/gtx1080 999999937 LL    80000   0.01%; 51002 us/it; ETA 590d 06:05; 2b4ab3e44fcbdc84
2021-03-27 16:03:07 asr3/gtx1080 999999937 LL    90000   0.01%; 50957 us/it; ETA 589d 17:20; b9c6eeca4e553904
2021-03-27 16:11:37 asr3/gtx1080 999999937 LL   100000   0.01%; 50972 us/it; ETA 589d 21:24; e776f4a0dcd3491d

CUDALucas V2.06beta on V100 (LL, no Jacobi check, unverified, ~6 months; interim residue extracted by ramgeis from an interim save file using a perl program provided by Kriesel) This run was preceded and followed by successful LL DC runs. However, it was not a paired run with interim residues cross-checked, so its reliability over the long run time is uncertain.
999999937 Iteration 110,000,000 Residue64 0x87ded9d3ab3e7f38
See also https://mersenneforum.org/showthread...859#post495859


The PRP equivalent is feasible marginally with Gpuowl on a Radeon VII, at ~0.5 years to complete; LL in Gpuowl on a Radeon VII is also possible but not recommended since it's only protected by an occasional Jacobi check's ~50% chance of detection of an error. Such long runs are likely to go wrong with undetected errors.
Gpuowl v6.2-e2ffe65 PRP on RX480:

2019-02-08 17:48:10 condorella/rx480 999999937       10000  0.00%; 72.02 ms/sq; ETA 833d 14:01; a30a0c45e9fb828c
2019-02-08 21:16:26 condorella/rx480 999999937      100000  0.01%; 74.38 ms/sq; ETA 860d 18:51; 3efc806b68d92b86

continuation on gpuowl v6.11-380-g79ea0cc on RX480, quite a speed improvement, 1.51X over v6.2, but still not recommended at ~1.56 years on an RX480:

2021-03-16 12:20:54 condorella/rx480 999999937 OK  1000000   0.10%; 49236 us/it; ETA 569d 07:00; e1b53ccf581f928b (check 22.08s)

1.25Gbit experiment
CUDALucas on GTX1080 wrote a save file and stopped upon request.

Using threads: square 512, splice 64.
Starting M1250000033 fft length = 73728K
    SIGINT caught, writing checkpoint. Estimated time spent so far: 12:18

A resumption produced

|   Date     Time    |   Test Num     Iter        Residue        |    FFT   Error     ms/It     Time  |       ETA      Done   |
|  Mar 28  13:21:04  | M1250000033     10000  0xf40e592716cd5c7a  | 73728K  0.12500  75.3094   11.14s  | 1084:17:26:53   0.00%  |

This residue is unverified and conflicts with the one produced by gpuowl v6.11-380 on a Radeon VII gpu:

2021-03-30 18:45:38 gpuowl v6.11-380-g79ea0cc
2021-03-30 18:45:38 config: -user kriesel -cpu asr2/radeonvii4 -d 4 -use NO_ASM -maxAlloc 15000 -cleanup -block 1000 -log 10000
2021-03-30 18:45:38 device 4, unique id ''
2021-03-30 18:45:38 asr2/radeonvii4 1250000033 FFT: 72M 4K:9:1K (16.56 bpw)
2021-03-30 18:45:38 asr2/radeonvii4 Expected maximum carry32: 6CC80000
2021-03-30 18:45:49 asr2/radeonvii4 OpenCL args "-DEXP=1250000033u -DWIDTH=4096u -DSMALL_HEIGHT=1024u -DMIDDLE=9u -DPM1=0 -DAMDGPU=1 -DWEIGHT_STEP_MINUS_1=0xb.819f9530b86cp-5 -DIWEIGHT_STEP_MINUS_1=-0x8.76945b26097f8p-5 -DNO_ASM=1  -cl-unsafe-math-optimizations -cl-std=CL2.0 -cl-finite-math-only "
2021-03-30 18:45:54 asr2/radeonvii4 OpenCL compilation in 4.79 s
2021-03-30 18:46:00 asr2/radeonvii4 1250000033 LL        0 loaded: 0000000000000004
2021-03-30 18:48:54 asr2/radeonvii4 1250000033 LL    10000   0.00%; 17459 us/it; ETA 252d 13:58; 5a7f1ee464e7c654
2021-03-30 18:51:48 asr2/radeonvii4 1250000033 LL    20000   0.00%; 17428 us/it; ETA 252d 03:27; 1bdc8fcb27f794f5
2021-03-30 18:52:26 asr2/radeonvii4 Stopping, please wait..
2021-03-30 18:52:29 asr2/radeonvii4 1250000033 LL    22000   0.00%; 20088 us/it; ETA 290d 14:53; b2fc2c1ace615898
2021-03-30 18:52:29 asr2/radeonvii4 waiting for the Jacobi check to finish..
2021-03-30 19:11:16 asr2/radeonvii4 1250000033 OK    22000 (jacobi == -1)

1.4xGbit experiments
These were performed after the 2Gbit, 1.5Gbit, and 1.25Gbit experiments, to determine the approximate transition point for CUDALucas. The following are in exponent order, not chronological order.

1.40G

Using threads: square 1024, splice 32.
Starting M1400000197 fft length = 81920K
|   Date     Time    |   Test Num     Iter        Residue        |    FFT   Error     ms/It     Time  |       ETA      Done   |
|  Mar 27  09:32:44  | M1400000197     10000  0x3064425057775e29  | 81920K  0.17188  85.1755  851.75s  | 1380:03:35:56   0.00%  |
exit at Sat 03/27/2021  9:47:05.96

1.43G is about the highest that would run in CUDALucas on the GTX1080 long enough to produce any printed interim residues; 1.45Gbit and higher failed early.

Starting M1430000027 fft length = 81920K
|   Date     Time    |   Test Num     Iter        Residue        |    FFT   Error     ms/It     Time  |       ETA      Done   |
|  Mar 27  11:28:40  | M1430000027     10000  0xa58244f8b0e73cde  | 81920K  0.26563  85.7222  857.22s  | 1418:18:31:51   0.00%  |
|  Mar 27  11:42:59  | M1430000027     20000  0x210de9223d1c0ca0  | 81920K  0.29688  85.9697  859.69s  | 1420:19:27:07   0.00%  |
|  Mar 27  11:57:19  | M1430000027     30000  0xf17cde78a6256ad9  | 81920K  0.28125  85.9585  859.58s  | 1421:10:07:19   0.00%  |
|  Mar 27  12:11:38  | M1430000027     40000  0x6863e2b56e6e5d89  | 81920K  0.28125  85.9554  859.55s  | 1421:17:01:24   0.00%  |
exit at Sat 03/27/2021 12:26:00.77

Compare the above 10000 iter residue to gpuowl v6.11-380's Jacobi checked output 19b5110e4fd08ef6

At 1.44G CUDALucas struggled and crashed

Using threads: square 32, splice 32.
Starting M1440000083 fft length = 82944K
Round off error at iteration = 100, err = 0.5 > 0.35, fft = 82944K.
Restarting from last checkpoint to see if the error is repeatable.

Using threads: square 32, splice 32.
Starting M1440000083 fft length = 82944K
Round off error at iteration = 100, err = 0.5 > 0.35, fft = 82944K.
The error persists.
Trying a larger fft until the next checkpoint.

Using threads: square 32, splice 128.
Starting M1440000083 fft length = 84672K

1.45G just crashed CUDALucas.

Using threads: square 32, splice 32.
Starting M1450000043 fft length = 82944K

The 100M steps in test exponent sometimes correspond to different expected fft lengths.

1.5Gbit experiments
interim residues
gpuowl v4.6 PRP on RX480 (not recommended, at estimated 7.8 years to completion; also moot since it has a known small factor) B1 bounds 0, so presumably base = 3.
2018-11-01 16:35:37 condorella-rx480 10000/1500000041 [ 0.00%], 164.51 ms/it [163.50, 174.58]; ETA 2856d 01:42;
CUDALucas v2.06 fails on 1,500,000,043 with an error similar to that for 2Gbit (see below).

Using threads: square 512, splice 128.
Starting M1500000043 fft length = 86400K
Round off error at iteration = 100, err = 0.5 > 0.35, fft = 86400K.
Something is wrong! Quitting.

2Gbit experiments
https://www.mersenne.ca/exponent/2147483563 is theoretically just within range of CUDALucas. It would need a lot of TF and P-1 before a serious attempt at LL were worthwhile. Estimated time on a GTX1080 is several years to completion, and little or no chance of correct completion without the Jacobi check or better.
In initial attempts to obtain LL timing on a GTX 1080, CUDALucas halted repeatedly before producing any timing data. A few more attempts were made after tuning the application for the GPU on the system where it's currently installed. It initially selects a properly sized fft length, decides that's too large, goes to one too small, and has excessive round-off error, terminating, even if a proper fft length is specified on the command line.

The fft length 131072K is too large for exponent 2147483563, decreasing to 116640K
Using threads: square 1024, splice 64.
Starting M2147483563 fft length = 116640K
Round off error at iteration = 100, err = 0.5 > 0.35, fft = 116640K.
Something is wrong! Quitting.

The relevant fft lengths and benchmark timings are

Device              GeForce GTX 1080
Compatibility       6.1
clockRate (MHz)     1797
memClockRate (MHz)  5005

  fft    max exp  ms/iter
76832 1335757897  83.2664
81920 1422251777  83.9003
82944 1439645131  84.9825
84672 1468986017  91.3988
86400 1498314007  94.8596
93312 1615502269  96.7309
96768 1674025489  99.3981
98304 1700021251 103.3561
100352 1734668777 105.2496
102400 1769301077 110.3034
104976 1812840839 112.3060
110592 1907684153 113.6384
114688 1976791967 117.1181
115200 1985426669 124.0529
116640 2009707367 131.0044
131072 2147483647 131.6075

The issue was reproducible on 2,000,000,099 and some lower values.

Using threads: square 1024, splice 64.
Starting M2000000099 fft length = 115200K
Round off error at iteration = 100, err = 0.5 > 0.35, fft = 115200K.
Restarting from last checkpoint to see if the error is repeatable.

Using threads: square 1024, splice 64.
Starting M2000000099 fft length = 115200K
Round off error at iteration = 100, err = 0.5 > 0.35, fft = 115200K.
The error persists.
Trying a larger fft until the next checkpoint.

Using threads: square 1024, splice 64.
Starting M2000000099 fft length = 116640K
Round off error at iteration = 100, err = 0.5 > 0.35, fft = 116640K.
Something is wrong! Quitting.

P-1 stage 1 appears possible on gpuowl v6.11-380, at over 6 days on a Radeon VII:

2021-03-26 13:57:40 asr2/radeonvii4 2147483563 P1 B1=11000000, B2=600000000; 15869712 bits; starting at 0
2021-03-26 14:02:48 asr2/radeonvii4 saved
2021-03-26 14:03:21 asr2/radeonvii4 2147483563 P1    10000   0.06%; 34113 us/it; ETA 6d 06:17; d78cbf554970d8c1

P-1 stage 2 is unlikely to work, since gpu ram of 16 GB is barely adequate (-maxalloc 15000) for exponents near 1G.

https://www.mersenne.ca/exponent/2147483743 would need a lot of TF and P-1 first, before a serious attempt at PRP testing was made. This is far beyond the capability of CUDAPm1, and requires a 16GB GPU in gpuowl. The LL is slightly beyond the nominal range of CUDALucas, but PRP is theoretically within reach of gpuowl or Mlucas and TF is feasible in Mfaktx. Projected PRP time on a Radeon VII is 2.63 years, not recommended. Gpuowl V6.11-380 PRP timings:

2021-03-26 12:43:44 asr2/radeonvii4 2147483743 OK    20000   0.00%; 38636 us/it; ETA 960d 07:01; 304a08968e896b20 (check 23.14s)
2021-03-26 13:36:46 asr2/radeonvii4 2147483743 OK   100000   0.00%;  38636 us/it; ETA 960d 05:56; 0637bf5d33611a9d (check 22.85s)

gigadigit ~3.3gigabit experiments
There may be some naive interest in attempting these exponents, since they would potentially qualify for the largest EFF prize. Sufficient TF is feasible with mfaktx and a fast gpu or a lot of patience. One exponent has been trial factored to sufficient bit depth.

There is currently no GIMPS software suitable for P-1 factoring gigadigit exponents preparatory to a primality test attempt. As an experiment a copy of Gpuowl v6.11-219 was modified to raise the P-1 exponent limit as a function of fft length, sufficiently to permit P-1 attempts on gigabit exponents with its highest fft length. It was tested at a much smaller exponent and fft length, near the expanded P-1 bits/word limit, and failed to find a known factor, as one might expect. (George has identified a factor of 3 as a reason for slightly lower usable bits/word in the fft lengths for P-1.) P-1 factoring of gigadigit exponents will require a larger fft length than the 192M largest ever offered in Gpuowl. Estimated runtime for a gigadigit stage 1 P-1 run on a Radeon VII would be ~1-2 months per exponent. That is impractically long without robust error correction such as introduced in Gpuowl V7.2 P-1.

Mlucas and a select few versions of gpuowl could run the primality tests. Primality test duration is too long on any available consumer-grade hardware. There are severe reliability issues with very long LL tests.

Gpuowl v6.5-84 PRP3 (not recommended at an estimated 6.3 years on a Radeon VII; lacks PRP proof capability)
M3,321,928,097 1,000,000,001 decimal digits (This exponent is moot, as it has multiple known small factors)

FFT 196608K: Width 512x8, Height 256x8, Middle 12; 16.50 bits/word
2020-05-24 17:21:43 radeonvii3 3,321,928,097       20000  0.00%; 60053 us/sq; ETA 2308d 21:54; 1a05f0ca51fb8e7a
2020-05-24 18:41:56 radeonvii3 3,321,928,097      100000  0.00%; 60122 us/sq; ETA 2311d 12:24; 5b2cb77f57840bcc

A better test would have been a well trial factored OBD candidate such as 3321928171.


The decline in bits/word is 1.09 over the range above, or about 0.838 bits/word per factor of ten on exponent.


Top of this thread https://www.mersenneforum.org/showthread.php?t=24003
Top of reference tree: https://www.mersenneforum.org/showpo...22&postcount=1

[kriesel]

V1:
"An attempt at documenting the file format is here:
https://github.com/preda/gpuowl/wiki/PRP-Proof-File-Spec"
(reproduced from https://mersenneforum.org/showpost.p...6&postcount=87)

(following first appeared as part of https://mersenneforum.org/showpost.p...8&postcount=95; edited for correctness and completeness here)
Header fields are expressed in 8-bit ASCII, 1 character per byte. Some header records are variable in length.

All header fields shown in the example are required. Format and order must be as described. Except for the first header line, the structure is descriptor, "=", value, newline character (0x0A). Numerical values are expressed as unsigned base-ten integers in ASCII. <variable> below means insert value of variable. Alpha characters in header descriptors shall be upper case.

First header record is file type: PRP PROOF\n
Second header record is version: VERSION=<version>\n
Third header record is hashsize: HASHSIZE=<hashsize>\n
Fourth header record is power: POWER=<power>\n
Fifth header record is number: NUMBER=<numbertype><exponent>\n

Only hashsize=64 is currently supported.
Proof power 6, 7, 8, 9 or 10 are allowed values. Power 8 is preferred.
Only Mersenne type numbers are currently supported, denoted by numbertype="M"

Following the header's last \n, residues B and Middle(0), Middle(1),...,Middle(power-2),Middle(power-1) follow consecutively, in that order, without separator fields. Each residue is expressed as an unsigned binary multibyte integer, in least-significant-byte-first byte order, in a whole number of bytes, and its MSB is zero-extended. Byte count per residue is not present in the header, and is computed from the NUMBER field of the header. The residues' first byte position offsets are computable from header length, exponent, and which residue is sought.

Abridged example:

PRP PROOF
VERSION=1
HASHSIZE=64
POWER=9
NUMBER=M1257787

V2:
A wider range of proof powers is allowed. (Gpuowl v7.2-53 accepts 1-10.) Powers lower than 8 should only be used when necessary, such as on small computing devices such as Raspberry Pi, Intel compute sticks, and other systems with limited space for temporary files, or to make possible the proof generation of a run begun without proof generation. Computing-effort optimal (test, proof generation, server and certification combined) is around power 10. Use the maximum practical proof power to save total computation time.
Abridged example:

PRP PROOF
VERSION=2
HASHSIZE=64
POWER=4
NUMBER=M859433

Top of this thread https://www.mersenneforum.org/showthread.php?t=24003
Top of reference tree: https://www.mersenneforum.org/showpo...22&postcount=1