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Old 2007-05-04, 11:39   #1
Thomas11
 
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Feb 2003

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Default false composites with LLR

During my search for high weight k I found a severe problem with LLR:
At least for very small n (n<500) there is a chance for randomly generated "false composites".

More or less by accident I'd rerun an input file which had already been tested by LLR. Just out of curiosity I counted the numbers of primes in both output files and found that the counts did not match. By comparing the lresults.txt files I found the following discrepancy:

Code:
21375317535*2^47-1 is prime!  Time : 0.811 ms.
21375317535*2^54-1 is not prime.  LLR Res64: A1154AE6B7462DA9  Time : 0.684 ms.
21375317535*2^55-1 is not prime.  LLR Res64: 9380000000066E07  Time : 1.509 ms.
21375317535*2^61-1 is prime!  Time : 0.712 ms.
Code:
21375317535*2^47-1 is prime!  Time : 0.610 ms.
21375317535*2^54-1 is not prime.  LLR Res64: A1154AE6B7462DA9  Time : 0.648 ms.
21375317535*2^55-1 is prime!  Time : 0.669 ms.
21375317535*2^61-1 is prime!  Time : 0.722 ms.
Note the much larger time for the false composite in the first file.
Also note the "lots of zeros" pattern in the corresponding Res64 value.

And there was another one:
Code:
9934327416585*2^79-1 is not prime.  RES64: 000000000038AACA.  OLD64: 0000000000A
A005D  Time: 0.709 ms.
-----------------------------------------------------------------------------
9934327416585*2^79-1 is a probable prime.  Time: 0.686 ms.
9934327416585*2^79-1 is prime!  Time : 0.804 ms.
Inititally I thought of another "false prime", but a check by Pari confirmed the primality of both numbers. So my second thought was about a hardware error. However, memtest86 as well as a prime95 stress test didn't show any problems. Both, the LLR test as well as the PRP test in LLR are affected. We already had a similar problem in the beginning of the 15k project, but then only the SSE2 code was missing some primes in the n<250 range, and the false composites were reproducable.

However, this time it seems that the false composites turn out completely ramdomly, and there is almost no chance to reproduce this phenomenon for a specific number.
I dicided to prepare a large test bench containing 4,000,000 candidates (8000 k with n=1-500, completely unsieved), which took less than two hours on my 2GHz Athlon. It yielded 235278 primes (and prp's) in the output file. Then I used that output file as input again for another LLR run. This time 43 of the former primes turned out as composites:

Code:
666625245*2^45-1 is not prime.  LLR Res64: 7C9B9FFFFFFFFFFE  Time : 0.572 ms.
10446821385*2^51-1 is not prime.  LLR Res64: CCF7FFFFFE5B0199  Time : 0.659 ms.
13947568635*2^6-1 is not prime.  RES64: 000000AF3AD8BEFC.  OLD64: 0000006E05093F73  Time: 0.460 ms.
17604081495*2^151-1 is not prime.  RES64: FFFFFFFFFE0C567F.  OLD64: FFFFFFFFFA25037A  Time: 0.879 ms.
29069604135*2^109-1 is not prime.  LLR Res64: FFFFFFFFFED169A6  Time : 0.806 ms.
30067977225*2^16-1 is not prime.  RES64: 0002F51953756EB7.  OLD64: 0001DF1B0E574C23  Time: 0.558 ms.
30832176375*2^105-1 is not prime.  RES64: 0000000000011D4B.  OLD64: 00000000000357DE  Time: 0.747 ms.
33711967575*2^21-1 is not prime.  RES64: 00429AC4A46104EC.  OLD64: 00C7D04DED230EC1  Time: 0.639 ms.
34449023895*2^17-1 is not prime.  RES64: 000B7C0A0CBA0A9E.  OLD64: 00025ED497D21FD9  Time: 0.566 ms.
38994969585*2^96-1 is not prime.  LLR Res64: FFFFFFFFFE3CDC30  Time : 0.803 ms.
41855065395*2^50-1 is not prime.  LLR Res64: 933FFFFFFF2DD810  Time : 0.634 ms.
49489163295*2^109-1 is not prime.  LLR Res64: FFFFFFFFFE9C8644  Time : 0.812 ms.
52690713075*2^138-1 is not prime.  LLR Res64: 0000000000537817  Time : 0.916 ms.
54574705185*2^68-1 is not prime.  LLR Res64: 0000000000014259  Time : 0.738 ms.
58584259305*2^87-1 is not prime.  RES64: FFFFFFFFFFDA76C4.  OLD64: FFFFFFFFFF8F644B  Time: 0.694 ms.
63346678395*2^79-1 is not prime.  RES64: 0000000000022B6D.  OLD64: 0000000000068246  Time: 0.682 ms.
73924325475*2^109-1 is not prime.  RES64: FFFFFFFFFF890320.  OLD64: FFFFFFFFFE9B095E  Time: 0.825 ms.
80621946405*2^72-1 is not prime.  LLR Res64: 0000000000DA34B7  Time : 0.748 ms.
85597136625*2^104-1 is not prime.  LLR Res64: 0000000000019B0B  Time : 0.821 ms.
85686205605*2^64-1 is not prime.  LLR Res64: FFFFFFFFFFF32B0C  Time : 0.752 ms.
87029734935*2^83-1 is not prime.  LLR Res64: 0000000003288F4E  Time : 0.767 ms.
97886532315*2^14-1 is not prime.  RES64: 0002744E977511E9.  OLD64: 0001AA4C49A875B9  Time: 0.559 ms.
99754874505*2^13-1 is not prime.  RES64: 00023B171119AD55.  OLD64: 0000E2CE8DAAC7FE  Time: 0.546 ms.
104420708535*2^66-1 is not prime.  RES64: FFFFFFFFFFFABC36.  OLD64: FFFFFFFFFFF0349F  Time: 0.660 ms.
110557596435*2^92-1 is not prime.  RES64: FFFFFFFFFFFEC1C2.  OLD64: FFFFFFFFFFFC4544  Time: 0.803 ms.
114971118405*2^86-1 is not prime.  RES64: 00000000002640E5.  OLD64: 000000000072C2AE  Time: 0.723 ms.
118017194295*2^21-1 is not prime.  RES64: 0234601C7FB1089C.  OLD64: 032DD466583319D2  Time: 0.943 ms.
119183229165*2^11-1 is not prime.  RES64: 00005DBAA3A6019E.  OLD64: 00003B30ECEA9CD8  Time: 0.537 ms.
124751340285*2^40-1 is not prime.  LLR Res64: B608B9000011A093  Time : 0.575 ms.
127464319125*2^40-1 is not prime.  LLR Res64: 1F596D0000114087  Time : 0.684 ms.
130297851255*2^90-1 is not prime.  RES64: FFFFFFFFFFFDE3F1.  OLD64: FFFFFFFFFFF9ABD1  Time: 0.713 ms.
142982793525*2^39-1 is not prime.  LLR Res64: C53D2280000F612F  Time : 0.578 ms.
143902440825*2^1-1 is not prime.  RES64: 000000146A84C1FE.  OLD64: 0000003D3F8E45F7  Time: 1.085 ms.
149357253045*2^88-1 is not prime.  LLR Res64: 00000000000EB926  Time : 0.794 ms.
183608570145*2^124-1 is not prime.  RES64: FFFFFFFFFFD017DF.  OLD64: FFFFFFFFFF70479A  Time: 0.798 ms.
242481123885*2^26-1 is not prime.  RES64: 8849EAC8FFEDDCC1.  OLD64: B709BF124BC99641  Time: 0.570 ms.
334983317955*2^7-1 is not prime.  RES64: 0000083688CEFDFD.  OLD64: 000018A39A6CF9F4  Time: 0.515 ms.
349610450475*2^14-1 is not prime.  RES64: 0001E7B51664571F.  OLD64: 0005B71F432D055A  Time: 0.566 ms.
455843900655*2^2-1 is not prime.  RES64: 00000102B0FBC89D.  OLD64: 0000015F894BA619  Time: 0.469 ms.
513806395245*2^88-1 is not prime.  RES64: 00000000000223D3.  OLD64: 0000000000066B78  Time: 0.695 ms.
686256694695*2^79-1 is not prime.  LLR Res64: FFFFFFFFF9975C43  Time : 0.782 ms.
692050343985*2^78-1 is not prime.  LLR Res64: 00000000065AE795  Time : 0.797 ms.
1009734264825*2^26-1 is not prime.  RES64: 9F0A1B1534045B0C.  OLD64: 84577FFFD40D1123  Time: 0.592 ms.
All of them are confirmed prime by Pari.
Note the occurrence of the "FFFFFF" residuals besides the "000000" ones. And there are some others which do not show these patterns.

I repeated this retesting of the 235278 primes on several P4 machines too, using LLR 3.7.0 as well as 3.7.1. In each case it yielded about 40-45 "false composites", but each of the sets is different.

Having this in mind it wasn't a big surprise that checking the inial lresults.txt file for the occurrence of "000000" and "FFFFFF" revealed another 31 false composites.

Since the input file containing the 4 million tests is too big to be posted here (about 65MB) I attached a list of the 8000 k-values together with a perl script for generating the LLR input file:

Code:
make_input.pl k_list.txt > input.txt
Before running LLR on that file make sure that there is enough space (at least 500 MB) in the file system, since the lresults.txt gets really large. Once the LLR run has been finished (after about 1-2 hours, depending on your system) you may use the output.txt file as input again for another LLR run.

The ZIP file contains a second perl script (prep_pari.pl) which can be use to generate an pari/gp input file to check whether a composite is prime or not.
If you're running Linux then you may use something like the following for detecting false composites in the lresults.txt file:
Code:
grep "FFFFFF" lresults.txt > bad_list.txt
prep_pari.pl bad_list.txt > pari_input.txt
gp < pari_input.txt > pari_output.txt
So far it seems that only the small n are affected. And there is no simple rule to detect all false composites (only the "000000" and "FFFFFF" residuals). Perhaps, George and Jean may find the reason for this strange behaviour and solve this problem.

Edit: Note, that a "000000" does not automatically mean a false composite. For small values of n (e.g. n<10) the residuals usually contain lots of zeros...
Attached Files
File Type: zip llr_test.zip (44.6 KB, 143 views)

Last fiddled with by Thomas11 on 2007-05-04 at 11:49
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Old 2007-05-04, 12:33   #2
paulunderwood
 
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I am running the whole set of numbers generated by your Perl script. In the meantime here is one example:

From lresults.txt:

19030751025*2^101-1 is not prime. RES64: 000000000039C689. OLD64: 0000000000AD539A Time: 0.803 ms.

With PFGW:

paul@pp4:~/pfgw$ ./pfgw_ver_12_linux -q"19030751025*2^101-1" -tp
PFGW Version 1.2.0 for Pentium and compatibles [FFT v23.8]

Primality testing 19030751025*2^101-1 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling Brillhart-Lehmer-Selfridge with factored part 74.81%
19030751025*2^101-1 is prime! (0.0020s+0.0327s)

Rerunning LLR:

Starting Lucas Lehmer Riesel prime test of 19030751025*2^101-1
Using General Mode (Rational Base) : Mersenne fftlen = 32, Used fftlen = 32
V1 = 5 ; Computing U0...
V1 = 5 ; Computing U0...done.
Starting Lucas-Lehmer loop...
19030751025*2^101-1 is prime! Time : 34.379 ms.

Hopefully this does not affect the irrational base DWT calculations...

Last fiddled with by paulunderwood on 2007-05-04 at 12:37
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Old 2007-05-04, 14:36   #3
Thomas11
 
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Quote:
Originally Posted by Thomas11 View Post
In each case it yielded about 40-45 "false composites", but each of the sets is different.
Obviously I was wrong with that statement. In fact the sets of "false composites" are reproducable (when using an "all primes" input file). However, you'll get different sets for the SSE2 code and the non-SSE2 code (e.g. when disabling SSE2 by the use of "CpuSupportsSSE2=0" in the llr.ini). It did two additional tests on an Opteron (with and without SSE2) and got the same results as for the P4 (SSE2) and the Athlon (no SSE2), respectively.

The set of false composites for the P4/Opteron(with SSE2) is the following:
Code:
146110965*2^119-1 is not prime.  LLR Res64: FFFFFFFFFF8A6B45  Time : 1.443 ms.
2116838295*2^21-1 is not prime.  RES64: 000CF7B82D4FC4B4.  OLD64: 00075C0FA22F4E1B  Time: 0.881 ms.
2532474945*2^59-1 is not prime.  LLR Res64: A7FFFFFFFFFC9BAB  Time : 1.268 ms.
8866861575*2^37-1 is not prime.  LLR Res64: 5AB5279FF83FF665  Time : 1.067 ms.
10548249855*2^114-1 is not prime.  LLR Res64: 0000000000341E42  Time : 1.660 ms.
12744043455*2^25-1 is not prime.  RES64: 021B0272489B9368.  OLD64: 0061D1FF5BD2BA36  Time: 0.893 ms.
21344712675*2^70-1 is not prime.  LLR Res64: FFFFFFFFFFFF31F3  Time : 1.325 ms.
21371962755*2^99-1 is not prime.  LLR Res64: 0000000000CDC92A  Time : 1.569 ms.
21440002155*2^17-1 is not prime.  RES64: 0005756B4D665BC1.  OLD64: 00066468F75D1341  Time: 0.877 ms.
27211257645*2^79-1 is not prime.  LLR Res64: 00000000000A1A04  Time : 1.347 ms.
29645918445*2^42-1 is not prime.  LLR Res64: 2AE34FFFFFF6BA5C  Time : 1.219 ms.
34502279955*2^65-1 is not prime.  LLR Res64: FFFFFFFFFFE021D8  Time : 1.316 ms.
34947161535*2^90-1 is not prime.  LLR Res64: FFFFFFFFFFF8226C  Time : 1.408 ms.
39393823755*2^25-1 is not prime.  RES64: 080345FCC2DF493E.  OLD64: 05B1B5DE329DDBB8  Time: 0.895 ms.
45056520795*2^16-1 is not prime.  RES64: 00073EBD03E431B2.  OLD64: 0000C10F26F69515  Time: 0.899 ms.
48672677625*2^31-1 is not prime.  RES64: 492552100002D2E1.  OLD64: 30E0E6B3800878A1  Time: 0.914 ms.
68626201215*2^8-1 is not prime.  RES64: 00000D91D064CC4E.  OLD64: 000008C08FB966E9  Time: 0.776 ms.
69257213025*2^89-1 is not prime.  RES64: 00000000000FE033.  OLD64: 00000000002FA097  Time: 1.127 ms.
83747979315*2^164-1 is not prime.  RES64: FFFFFFFF00000000.  OLD64: FFFFFFFCFFFFFFFF  Time: 1.487 ms.
86905921245*2^83-1 is not prime.  LLR Res64: FFFFFFFFFE6B24E5  Time : 1.413 ms.
88749432915*2^64-1 is not prime.  RES64: 00000000000C6392.  OLD64: 0000000000252AB5  Time: 1.043 ms.
91938430155*2^112-1 is not prime.  LLR Res64: FFFFFFFFFFF40A70  Time : 1.592 ms.
93717793455*2^117-1 is not prime.  LLR Res64: 000000002EEDB9E5  Time : 1.675 ms.
98616136905*2^129-1 is not prime.  LLR Res64: 000000005931FD41  Time : 1.550 ms.
98848114365*2^38-1 is not prime.  LLR Res64: 301918FFFFD381CC  Time : 1.108 ms.
102467484405*2^91-1 is not prime.  LLR Res64: 000000000001575E  Time : 1.413 ms.
108581712525*2^66-1 is not prime.  RES64: FFFFFFFFFFFD77EE.  OLD64: FFFFFFFFFFF867C8  Time: 3.181 ms.
122169592545*2^24-1 is not prime.  RES64: 165F95659F8FFF82.  OLD64: 0A3B001B1CAFFE85  Time: 0.894 ms.
126360266175*2^4-1 is not prime.  RES64: 0000018A8B93D371.  OLD64: 000000F22DA6C272  Time: 0.749 ms.
136085318655*2^80-1 is not prime.  LLR Res64: FFFFFFFFFFFF7EBA  Time : 1.360 ms.
136129188195*2^51-1 is not prime.  LLR Res64: 2168000000204ED1  Time : 1.216 ms.
136690749195*2^103-1 is not prime.  RES64: FFFFFFFFFFFEFE9A.  OLD64: FFFFFFFFFFFCFBCB  Time: 1.160 ms.
140303937345*2^69-1 is not prime.  LLR Res64: 0000000003EB170F  Time : 1.745 ms.
141673060815*2^89-1 is not prime.  RES64: FFFFFFFFFFF83D36.  OLD64: FFFFFFFFFFE8B79F  Time: 1.128 ms.
249742686765*2^115-1 is not prime.  LLR Res64: 0000000023387AC0  Time : 1.479 ms.
279370430625*2^63-1 is not prime.  LLR Res64: FFFFFFFFFFF041DE  Time : 1.504 ms.
338164979295*2^57-1 is not prime.  LLR Res64: 4800000003405C24  Time : 1.276 ms.
548433554955*2^31-1 is not prime.  RES64: 853298E484027806.  OLD64: DE67DCA28C076811  Time: 0.906 ms.
557484506865*2^6-1 is not prime.  RES64: 000002A7AC26D473.  OLD64: 000007F704747D56  Time: 0.812 ms.
650832937755*2^64-1 is not prime.  LLR Res64: 00000000000360F7  Time : 1.363 ms.
845084472375*2^137-1 is not prime.  LLR Res64: 00000000000A6895  Time : 1.600 ms.
876707254995*2^60-1 is not prime.  LLR Res64: A0000000001410F2  Time : 1.658 ms.
927436552185*2^12-1 is not prime.  RES64: 000C3F5857708809.  OLD64: 0009C0182F52781A  Time: 0.870 ms.
1141813394865*2^21-1 is not prime.  RES64: 170C0DD186168408.  OLD64: 02ADD10526038C17  Time: 0.934 ms.
Now the question is: What's so special about these two sets of numbers?

Last fiddled with by Thomas11 on 2007-05-04 at 14:43
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Old 2007-05-04, 14:46   #4
paulunderwood
 
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Quote:
perl make_input.pl k_list.txt > input.txt
grep "FFFFFF" lresults.txt > bad_list.txt
perl prep_pari.pl bad_list.txt > pari_input.txt
gp < pari_input.txt > pari_output.txt
On my linux P4, results in pari_output.txt containing:

Quote:
15100664865*2^46-1 is prime!
25943940165*2^38-1 is prime!
32622298995*2^125-1 is prime!
41440213815*2^141-1 is prime!
43749718155*2^67-1 is prime!
45608725305*2^69-1 is prime!
48740395275*2^66-1 is prime!
88233019875*2^93-1 is not prime.
96533792355*2^55-1 is prime!
101023697775*2^103-1 is prime!
101406957795*2^101-1 is prime!
112453356015*2^62-1 is prime!
144272551995*2^89-1 is prime!
146649246315*2^67-1 is prime!
164454779775*2^69-1 is prime!
481476839415*2^374-1 is not prime.
1101302761845*2^60-1 is prime!
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Old 2007-05-04, 14:56   #5
Thomas11
 
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All but the two non-primes of Paul's list have been found on the Athlon in the initial LLR run (e.g. they haven't been "false composites" on the Athlon).

Obviously for the two non-primes the "FFFFFF" pattern in the residuals is quite okay...

Paul, what's your total number of primes in the output file (after the inital LLR run)?

BTW.: On the Athlon I had "only" 13 "FFFFFF" false composites...

Last fiddled with by Thomas11 on 2007-05-04 at 14:59
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Old 2007-05-04, 16:41   #6
Prime95
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Quote:
Originally Posted by Thomas11 View Post
Now the question is: What's so special about these two sets of numbers?
They're small. It's easier for me to multiply million bit numbers than 50-bit numbers.

I have some QA code that randomly picks numbers 500+ bit numbers to test. I'll modify it to test some tiny numbers to see if the problem is in the gwnum library.
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Old 2007-05-04, 17:05   #7
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I just tested 44 numbers from message #1 (there are 44 of them, not 43) and LLR found them all to be either primes or PRPs (when k>2^n). This is an old version of 2004.12.26 which is still the best on my Athlon (no SSE2). results.txt is attached.

In 2004 me and the others from prover's code L97 tested LLR on primes and composites with known residues found by a script written by David Broadhurst in Pari. That version of LLR was later released as ver. 3.6 if I remember correctly. Are you using the latest version 3.7.1? Maybe these small numbers are not handled properly in the latest version? Even so, I beleive large numbers (like those in Top-5000) are not affected.

Edit: All false composites from message #3 also found by LLR to be either primes or PRPs. The two composites from message #4 found to be composites but that's correct (they are not primes, confiremed by pfgw).
Attached Files
File Type: txt lresults.txt (3.8 KB, 126 views)

Last fiddled with by Kosmaj on 2007-05-04 at 17:15
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Old 2007-05-04, 21:00   #8
Thomas11
 
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Quote:
Originally Posted by Kosmaj View Post
I just tested 44 numbers from message #1 (there are 44 of them, not 43) and LLR found them all to be either primes or PRPs (when k>2^n). This is an old version of 2004.12.26 which is still the best on my Athlon (no SSE2). results.txt is attached.

...

Edit: All false composites from message #3 also found by LLR to be either primes or PRPs. The two composites from message #4 found to be composites but that's correct (they are not primes, confiremed by pfgw).
Of course, when testing just those sets of 44 or so numbers, they all turn out to be primes or PRPs. However, the problem occurs when you're testing a large number of candidates (e.g. millions) - then a small fraction of them may be wrong. It seems to depend on the test history, e.g. the tests which have been done before the "problematic" candidates. Perhaps some arrays are not completely erased before the next test is started. Or when a test uses a smaller FFT length than the test before.

So far it seems that both of the newer versions, LLR 3.7.0 and 3.7.1, are identical in that behaviour.

BTW.: I count only 43 numbers in post #1 ...

Last fiddled with by Thomas11 on 2007-05-04 at 21:50
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Old 2007-05-04, 21:17   #9
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Attached is an input file which contains the first 15000 primes (or PRPs) from my initial Athlon run. Just put it to your machine and run it through LLR (should take about a minute or so). Then check the output for "not prime". Depending on your cpu type you will find the first few "false composites" given in posts #1 or #3.

Then run those "composite" numbers through pari or LLR again...
Attached Files
File Type: zip test_input.zip (37.5 KB, 112 views)

Last fiddled with by Thomas11 on 2007-05-04 at 21:18
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Old 2007-05-04, 21:41   #10
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I just used the input file given in my latest post for some quick checks on the following LLR versions: 3.3, 3.5, 3.6.2, 3.7, and 3.7.1

I found that 3.6.2 (and perhaps 3.6), 3.7, 3.7.1 are producing exactly the same "false composites" (identical Res64's). And it seems, that the earlier versions (e.g. 3.5) do not show any "is not prime" in the output!

Back to LLR 3.5 ???

Last fiddled with by Thomas11 on 2007-05-04 at 21:42
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Old 2007-05-04, 22:01   #11
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On one of my Core2Duo machines running WindowsXP I've got following false composites:
Code:
973197225*2^159-1 is not prime.  LLR Res64: FFFFFFFFFFDCB1A7  Time : 1.607 ms.
4985029335*2^24-1 is not prime.  RES64: 01186BDFC6FF2370.  OLD64: 00F70099A6FD6A4F  Time: 1.006 ms.
6187790895*2^44-1 is not prime.  LLR Res64: B619800000163618  Time : 1.400 ms.
7469291115*2^115-1 is not prime.  LLR Res64: 0000000000126689  Time : 1.532 ms.
I have used input.txt supplied by Thomas11 and LLR version 3.7.1.
Running those 4 numbers alone through LLR reveals all af them to be primes... I really hope that this affects only some small values of n... which I test with PFGW anyway...

Last fiddled with by Cruelty on 2007-05-04 at 22:11
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