Primes and Strong Pseudoprimes of the form xy + yx

Many moons ago in a previous life, I was surprised to receive an email out of the blue which claimed that there are only a finite number of non-trivial values of x and y such that xy + yx  is prime. By "non-trivial" I mean, of course, that we discount (x,y) = (p-1,1) where p is prime and we assume for convenience that x>y.

It's straightforward to put a few conditions on x and y which must be met for xy + yx to be prime. For a start, they must be of opposite parity and be co-prime (i.e. have no factors in common). These conditions only remove a few candidates, and it was not obvious to me that all but a finite number of the remainder had to be composite. I wrote a little search program and found a few dozen pairs which were strong pseudoprimes. The smallest of these, those up to 400 decimal digits or so, were proved prime with a number of different programs, principally UBASIC's APRT-CL test and François Morain's ECPP.

A couple of years later, I completed the search for all x < 1030, and proved several more candidates prime, including (405,364) and (406,289) which have more than a thousand digits each. There the matter rested until early 2001. I still don't have a clue as to whether the number of primes of this form is finite or infinite, but if forced to guess I would go for infinite.

Interest was re-raised in February 2001 when a few people started searching again. A number of them, especially Greg Childers and Andrey Kulsha, have been particularly effective in finding new pseudoprimes and proving some of them to be prime. Andrey holds the record for the largest pseudoprime yet discovered of this form. It has 87201 digits. The results of all the searchers are summarized in the table below. The smaller values, especially those which have x < 1000, have doubtless been found many times by many other researchers. The last entry with x<1000 was completed on 28 March 2002.

Note added 16 November 2004: I thought that the tables were complete to x<1000 but Andrey Kulsha discovered two PRPs which had been omitted and has since proved them prime. This shows the importance of double checking the computations! Andrey has now completed the double checking as far as x=3500 and found no further errors. Please contact me if you have any other corrections or additions to make to either table on this page.

More recently still, it was realized that numbers of this form are ideal test cases for general purpose primality proving programs. They have a simple algebraic description but no obvious cyclotomic properties which special purpose algorithms can exploit. There seems to be a reasonably high density of primes of this form and so there is a reasonable number of test cases to use. Finally, there seems to be a good distribution of small, medium and large sized candidates in the tables below.

I volunteer to co-ordinate any further searches and to keep the following tables as up to date as I can. I strongly suggest that searches be performed in order of increasing x values, and that all values of y be tested for each x in the range. It keeps things neat and tidy, and makes my task easier 8-)

Mark Rodenkirch has written a program to sieve out candidates which are divisible by small primes, so reducing the number which need to be subjected to an expensive PRP test.

Raffi Chaglassian has already searched the range 7500 < x <8000 for 100 < y < 1001. Anatoly Selevich has completed searching the range 8000 < x < 20000 for y < 1001, Andrey Kulsha has completed the range 20000 < x < 100000 for y< 11 and Anatoly Selevich is searching the range 7500 < x <20001 for 1000 < y< 2001 so, to avoid duplicating their work, you should start searches with y = 2001 or y = 1001 or y = 11 as appropriate.

Andrey is also co-ordinating a project to find the factorization of "small" composite values of  xy + yx . He's produced a very professional looking site, and I recommend that you take a look.

 

Ranges being searched
Range xmin Range xmax Searched by When reserved When completed Number PRPs Number primes
             
3 1030 Paul Leyland 1994 1997   88
1031 1050 Paul Leyland 2001-June-4 2001-June-4   4
1051 1500 Andrey Kulsha 2001-June-3 2001-June-23 13 28
1501 1700 Paul Leyland 2001-June-5 2001-June-11 15 3
1701 2000 Christ van Willegen 2001-June-5 2001-June-9 12 11
2001 2200 Greg Childers 2001-June-7 2001-June-9 12 7
2201 2400 Greg Childers 2001-June-9 2001-June-12 15 2
2401 2600 Peter Liaskovsky 2001-August-31 2001-September-3 17 5
2601 2700 Paul Leyland 2001-July-24 2001-August-8 10 3
2701 2800 Greg Childers 2001-July-30 2001-July-31 5 2
2801 3000 Greg Childers 2001-July-31 2001-August-5 13 1
3001 3050 Sander Hoogendoorn 2001-June-14 2001-June-19 2  
3051 3100 Peter Liaskovsky 2001-September-3 2001-September-5 4 2
3101 3200 Alexander Kuzmich 2001-September-16 2001-October-6 12  
3201 3379 Leonid Muraviov 2001-November-28 2002-March-28 11 1
3380 3500 Andrey Kulsha 2002-April-19 2002-May-19 7  
3501 4400 Paul Leyland 2001-December-21 2002-June-27 63 3
4401 4500 Greg Childers 2002-July-13 2002-August-13 8 1
4501 5000 Mark Rodenkirch 2002-September-7 2002-November-19 42  
5001 5100 Paul Leyland 2003-June-13 2003-June-23 11  
5101 5200 Paul Leyland 2003-June-23 2003-July-16 5  
5201 5500 Greg Childers 2003-July-21 2003-September-17 17  
5501 7500 Anatoly Selevich 2003-June-25 2004-August-7 154  
7501 20000 Available for y>2000     110 1
20001 100000 Available for y>10     9  
100001 Large Available -      
     
Total known so far: 567 162

The table below lists all the proven primes and strong pseudoprimes of the form xy + yx known to me at 6th October 2006. Greg Childers informs me that certificates for the ones he has proved prime are available at http://www.pa.uky.edu/~childers/certs/ and Christ van Willegen's certificates are at http://home.iae.nl/users/richtofe/certs.

Primes and Strong Pseudoprimes
x y Decimal Digits Finder of Pseudoprime Prover Date Proved Method/Program
             
3 2 2 Paul Leyland Paul Leyland Before 1998 table lookup
9 2 3 Paul Leyland Paul Leyland Before 1998 trial division
15 2 5 Paul Leyland Paul Leyland Before 1998 trial division
21 2 7 Paul Leyland Paul Leyland Before 1998 trial division
24 5 17 Paul Leyland Paul Leyland Before 1998 APRT-CL
32 15 38 Paul Leyland Paul Leyland Before 1998 APRT-CL
33 2 10 Paul Leyland Paul Leyland Before 1998 APRT-CL
38 33 58 Paul Leyland Paul Leyland Before 1998 APRT-CL
54 7 46 Paul Leyland Paul Leyland Before 1998 APRT-CL
56 3 27 Paul Leyland Paul Leyland Before 1998 APRT-CL
68 21 90 Paul Leyland Paul Leyland Before 1998 APRT-CL
69 8 63 Paul Leyland Paul Leyland Before 1998 APRT-CL
75 34 115 Paul Leyland Paul Leyland Before 1998 APRT-CL
76 9 73 Paul Leyland Paul Leyland Before 1998 APRT-CL
81 80 155 Paul Leyland Paul Leyland Before 1998 APRT-CL
87 56 153 Paul Leyland Paul Leyland Before 1998 APRT-CL
114 67 209 Paul Leyland Paul Leyland Before 1998 APRT-CL
114 97 227 Paul Leyland Paul Leyland Before 1998 APRT-CL
122 9 117 Paul Leyland Paul Leyland Before 1998 APRT-CL
135 32 204 Paul Leyland Paul Leyland Before 1998 APRT-CL
144 65 262 Paul Leyland Paul Leyland Before 1998 APRT-CL
158 45 262 Paul Leyland Paul Leyland Before 1998 APRT-CL
160 133 340 Paul Leyland Paul Leyland Before 1998 ECPP
171 98 341 Paul Leyland Paul Leyland Before 1998 ECPP
185 36 288 Paul Leyland Paul Leyland Before 1998 APRT-CL
206 51 352 Paul Leyland Paul Leyland Before 1998 ECPP
214 157 470 Paul Leyland Paul Leyland Before 1998 ECPP
215 76 405 Paul Leyland Paul Leyland Before 1998 ECPP
235 214 548 Paul Leyland Paul Leyland Before 1998 ECPP
237 200 546 Paul Leyland Paul Leyland Before 1998 ECPP
248 87 482 Paul Leyland Paul Leyland Before 1998 ECPP
318 91 623 Paul Leyland Paul Leyland Before 1998 ECPP
318 247 761 Paul Leyland Paul Leyland Before 1998 ECPP
322 111 659 Paul Leyland Paul Leyland Before 1998 ECPP
333 122 695 Paul Leyland Paul Leyland Before 1998 ECPP
343 342 870 Paul Leyland Paul Leyland Before 1998 ECPP
357 20 465 Paul Leyland Paul Leyland Before 1998 ECPP
387 142 833 Paul Leyland Paul Leyland Before 1998 ECPP
405 364 1038 Paul Leyland Paul Leyland Before 1998 ECPP
406 289 1000 Paul Leyland Paul Leyland Before 1998 ECPP
422 9 403 Paul Leyland Paul Leyland Before 1998 ECPP
425 184 963 Paul Leyland Paul Leyland Before 1998 ECPP
435 98 867 Paul Leyland Andrey Kulsha 2001-June-9 Titanix 2.1.0 (ECPP)
436 423 1146 Paul Leyland Andrey Kulsha 2001-June-11 Titanix 2.1.0 (ECPP)
444 365 1138 Paul Leyland Andrey Kulsha 2001-July-3 Titanix 2.1.0 (ECPP)
471 20 613 Paul Leyland Andrey Kulsha 2001-June-4 Titanix 2.1.0 (ECPP)
477 56 834 Paul Leyland Andrey Kulsha 2001-June-6 Titanix 2.1.0 (ECPP)
488 441 1291 Paul Leyland Christ van Willegen 2001-June-10 Titanix 2.1.0 (ECPP)
510 329 1284 Paul Leyland Christ van Willegen 2001-June-10 Titanix 2.1.0 (ECPP)
519 8 469 Paul Leyland Andrey Kulsha 2001-June-3 Titanix 2.1.0 (ECPP)
545 234 1292 Paul Leyland Christ van Willegen 2001-June-10 Titanix 2.1.0 (ECPP)
557 234 1320 Paul Leyland Christ van Willegen 2001-June-13 Titanix 2.1.0 (ECPP)
580 157 1274 Paul Leyland Christ van Willegen 2001-June-13 Titanix 2.1.0 (ECPP)
590 291 1454 Paul Leyland Christ van Willegen 2001-June-13 Titanix 2.1.0 (ECPP)
590 513 1599 Paul Leyland Christ van Willegen 2001-June-25 Titanix 2.1.0 (ECPP)
636 91 1246 Paul Leyland Christ van Willegen 2001-June-25 Titanix 2.1.0 (ECPP)
648 379 1671 Paul Leyland Christ van Willegen 2001-June-25 Titanix 2.1.0 (ECPP)
663 98 1321 Paul Leyland Christ van Willegen 2001-June-4 Titanix 2.1.0 (ECPP)
675 464 1800 Paul Leyland François Morain 2001-June-22 ECPP-V11.0.5alpha
681 518 1849 Paul Leyland François Morain 2001-July-2 ECPP-V11.0.5alpha
714 325 1794 Paul Leyland François Morain 2001-July-2 ECPP-V11.0.5alpha
717 32 1080 Paul Leyland Andrey Kulsha 2001-June-13 Titanix 2.1.0 (ECPP)
733 654 2064 Paul Leyland Greg Childers 2001-June-16 Titanix 2.1.0 (ECPP)
734 87 1424 Paul Leyland Greg Childers 2001-June-11 Titanix 2.1.0 (ECPP)
747 634 2094 Paul Leyland Christ van Willegen 2001-August-8 Titanix 2.1.0 + Primo 0.1.0 (ECPP)
773 34 1184 Paul Leyland Andrey Kulsha 2001-July-2 Titanix 2.1.0 (ECPP)
782 21 1034 Paul Leyland Andrey Kulsha 2001-June-27 Titanix 2.1.0 (ECPP)
790 443 2091 Paul Leyland Greg Childers 2001-June-19 Titanix 2.1.0 (ECPP)
790 731 2263 Paul Leyland Greg Childers 2001-July-3 Titanix 2.1.0 (ECPP)
795 544 2175 Paul Leyland Greg Childers 2001-July-9 Titanix 2.1.0 (ECPP)
798 191 1821 Paul Leyland Greg Childers 2001-June-23 Titanix 2.1.0 (ECPP)
806 441 2132 Paul Leyland Greg Childers 2001-July-16 Titanix 2.1.0 (ECPP)
815 298 2017 Paul Leyland François Morain 2001-July-16 ECPP-V11.0.5alpha
849 848 2487 Paul Leyland Greg Childers 2001-August-5

Titanix 2.1.0 + Primo 0.1.0 (ECPP)

872 309 2172 Paul Leyland Greg Childers 2001-July-31 Titanix 2.1.0 (ECPP)
883 394 2292 Paul Leyland François Morain 2001-July-27 ECPP-V11.0.5alpha
900 881 2651 Andrey Kulsha Andrey Kulsha 2004-November-16 Primo 2.2.0b5
903 278 2207 Paul Leyland François Morain 2001-August-25 ECPP-V11.0.5alpha
903 376 2326 Paul Leyland François Morain 2001-August-28 ECPP-V11.0.5alpha
907 694 2578 Paul Leyland François Morain 2001-September-3 ECPP-V11.0.5alpha
927 68 1699 Paul Leyland Peter Liaskovsky 2001-June-29 Titanix 2.1.0 (ECPP)
942 559 2589 Paul Leyland Christ van Willegen 2001-November-13 Primo 1.1.0 (ECPP)
958 373 2464 Andrey Kulsha Andrey Kulsha 2004-November-19 Primo 2.2.0b5
973 342 2466 Paul Leyland Greg Childers 2001-August-10

Primo 0.1.0 (ECPP)

983 54 1703 Paul Leyland Greg Childers 2001-June-16 Titanix 2.1.0 (ECPP)
988 521 2685 Paul Leyland Greg Childers 2001-June-29 Titanix 2.1.0 (ECPP)
991 942 2948 Paul Leyland François Morain 2002-March-28 ECPP-V11.0.5alpha
1015 384 2624 Paul Leyland Greg Childers 2001-September-27

Primo 1.1.0b (ECPP)

1036 5 725 Paul Leyland Andrey Kulsha 2001-June-5 Titanix 2.1.0 (ECPP)
1036 397 2693 Paul Leyland François Morain 2002-June-11 ECPP-V11.0.5alpha
1040 453 2763 Paul Leyland François Morain 2001-December-11 ECPP-V11.0.5alpha
1044 95 2065 Paul Leyland Peter Liaskovsky 2001-June-29 Titanix 2.1.0 (ECPP)
1052 927 3122 Paul Leyland Peter Liaskovsky 2001-November-14 Primo 1.1.0 (ECPP)
1054 743 3027 Andrey Kulsha François Morain 2002-April-22 ECPP-V11.0.5alpha
1071 752 3081 Andrey Kulsha François Morain 2002-May-17 ECPP-V11.0.5alpha
1072 491 2885 Andrey Kulsha Peter Liaskovsky 2001-September-23 Primo 1.0 (ECPP)
1080 569 2976 Andrey Kulsha François Morain 2002-June-29 ECPP-V11.0.5alpha
1131 920 3353 Andrey Kulsha François Morain 2002-August-19 ECPP-V11.0.5alpha
1132 425 2976 Andrey Kulsha François Morain 2002-March-1 ECPP-V11.0.5alpha
1139 666 3216 Andrey Kulsha François Morain 2002-August-16 ECPP-V11.0.5alpha
1143 242 2725 Andrey Kulsha Greg Childers 2001-October-14

Primo 1.1.0 (ECPP)

1148 321 2878 Andrey Kulsha François Morain 2002-January-8 ECPP-V11.0.5alpha
1154 1095 3508 Andrey Kulsha François Morain 2002-November-2 ECPP-V11.0.5alpha
1156 61 2064 Andrey Kulsha Peter Liaskovsky 2001-June-19 Titanix 2.1.0 (ECPP)
1158 101 2322 Andrey Kulsha Greg Childers 2001-August-9

Primo 0.1.0 (ECPP)

1158 751 3330 Andrey Kulsha Paul Leyland 2002-October-22 Moxham's prime-gmp 1.13
1195 1194 3678 Henri Lifchitz François Morain 2003-February-4 ECPP-V11.0.5alpha
1218 673 3445 Andrey Kulsha François Morain 2002-September-23 ECPP-V11.0.5alpha
1242 653 3497 Andrey Kulsha François Morain 2003-March-13 ECPP-V11.0.5alpha
1263 608 3517 Andrey Kulsha François Morain 2004-November-22 fastEcpp
1263 740 3624 Andrey Kulsha François Morain 2003-January-2 ECPP-V11.0.5alpha
1284 655 3617 Andrey Kulsha François Morain 2004-November-24 mpifastecpp
1293 92 2540 Andrey Kulsha Greg Childers 2001-August-31

Primo 0.2.0 (ECPP)

1298 45 2146 Andrey Kulsha Peter Liaskovsky 2001-June-24 Titanix 2.1.0 (ECPP)
1298 747 3730 Andrey Kulsha François Morain 2004-December-3 mpifastecpp
1305 988 3909 Andrey Kulsha François Morain 2005-January-21 fastECPP
1319 988 3951 Andrey Kulsha François Morain 2005-January-21 fastECPP
1325 1066 4012 Andrey Kulsha François Morain 2003-May-9 ECCP-11.0.5delta
1337 394 3471 Andrey Kulsha Décio Luiz Gazzoni Filho 2004-February-27 Primo 2.2.0 beta 3 (ECPP)
1358 831 3965 Andrey Kulsha Nathan Russell 2003-May-2 Primo 2.0.1 (ECPP)
1384 345 3513 Andrey Kulsha      
1388 621 3877 Andrey Kulsha      
1395 1232 4312 Andrey Kulsha      
1401 1040 4227 Andrey Kulsha      
1405 1188 4321 Andrey Kulsha      
1419 1106 4320 Andrey Kulsha      
1425 578 3936 Andrey Kulsha      
1442 999 4326 Andrey Kulsha      
1445 288 3554 Andrey Kulsha      
1451 1410 4570 Andrey Kulsha      
1476 931 4383 Andrey Kulsha      
1477 730 4230 Andrey Kulsha      
1491 968 4452 Andrey Kulsha      
1504 985 4503 Paul Leyland      
1507 1336 4711 Paul Leyland      
1527 50 2595 Paul Leyland Nuutti Kuosa 2001-August-27 Primo 0.2.0 (ECPP)
1528 77 2883 Paul Leyland François Morain 2002-February-1 ECPP-V11.0.5alpha
1545 782 4471 Paul Leyland      
1545 1394 4858 Paul Leyland      
1557 788 4510 Paul Leyland      
1568 1287 4876 Paul Leyland      
1576 1273 4894 Paul Leyland      
1588 721 4539 Paul Leyland      
1588 1047 4796 Paul Leyland      
1610 993 4826 Paul Leyland      
1616 633 4528 Paul Leyland      
1629 476 4362 Paul Leyland      
1632 775 4716 Paul Leyland      
1641 1298 5109 Paul Leyland François Morain 2004-August-23 ECPP-V11.05alpha
1667 252 4004 Paul Leyland      
1693 414 4431 Paul Leyland      
1723 1716 5574 Christ van Willegen Christ van Willegen 2003-November-3 Primo 2.2.0 beta 1 (ECPP)
1757 76 3305 Christ van Willegen Christ van Willegen 2001-December-11 Primo 1.1.0 (ECPP)
1774 723 5073 Christ van Willegen Christ van Willegen 2003-February-18 Primo 2.0.0 beta 3 (ECPP)
1785 596 4954 Christ van Willegen      
1792 1047 5412 Christ van Willegen François Morain 2004-September-08 ECPP-V11.0.5alpha
1813 688 5145 Christ van Willegen François Morain 2004-September-07 ECPP-V11.0.5alpha
1815 1088 5512 Christ van Willegen François Morain 2004-September-12 ECPP-V11.0.5alpha
1821 92 3577 Christ van Willegen      
1838 711 5242 Christ van Willegen François Morain 2004-September-08 ECPP-V11.0.5alpha
1842 1219 5685 Christ van Willegen      
1851 158 4070 Christ van Willegen      
1862 1485 5906 Christ van Willegen      
1869 482 5015 Christ van Willegen François Morain 2003-June-6 mpiecpp
1874 333 4728 Christ van Willegen      
1898 1041 5728 Christ van Willegen Christ van Willegen 2004-December-17 PRIMO 2.2.0 beta 4
1902 1757 6172 Christ van Willegen      
1915 1554 6112 Christ van Willegen      
1940 1911 6366 Christ van Willegen François Morain 2004-November-8 ECPP-V11.0.5alpha
1943 1396 6111 Christ van Willegen      
1946 1881 6372 Christ van Willegen François Morain 2004-September-14 ECPP-V11.0.5alpha
1956 713 5581 Christ van Willegen      
1982 711 5653 Christ van Willegen      
1991 1378 6251 Christ van Willegen      
2007 2 605 Rob Binnekamp Andrey Kulsha 2001-June-5 Titanix 2.1.0 (ECPP)
2007 1334 6273 Greg Childers François Morain 2004-September-17 mpifastecpp
2054 1137 6277 Greg Childers François Morain 2004-December-1 mpifastecpp
2066 459 5500 Greg Childers      
2067 1382 6492 Greg Childers François Morain 2004-August-23 ECPP-V11.05alpha
2075 1672 6689 Greg Childers      
2092 1803 6812 Greg Childers      
2095 988 6275 Greg Childers      
2103 950 6263 Greg Childers      
2105 424 5531 Greg Childers      
2115 802 6143 Greg Childers François Morain 2005-March-1 fastECPP
2127 2 641 Rob Binnekamp Andrey Kulsha 2001-June-5 Titanix 2.1.0 (ECPP)
2132 303 5291 Greg Childers      
2133 1474 6759 Greg Childers      
2151 356 5489 Greg Childers      
2170 879 6389 Greg Childers      
2172 371 5581 Greg Childers      
2177 580 6016 Greg Childers François Morain 2003-June-6 mpifastecpp
2184 295 5395 Greg Childers      
2240 87 4345 Greg Childers      
2276 141 4892 Greg Childers      
2278 537 6219 Greg Childers      
2284 1985 7533 Greg Childers      
2300 9 2195 Andrey Kulsha Greg Childers 2001-August-7 Primo 0.1.0 (ECPP)
2305 1374 7234 Greg Childers      
2317 1354 7256 Greg Childers      
2322 337 5870 Greg Childers      
2325 596 6453 Greg Childers      
2328 923 6903 Greg Childers      
2337 338 5911 Greg Childers      
2343 962 6990 Greg Childers      
2349 1772 7631 Greg Childers      
2361 662 6661 Greg Childers      
2362 1363 7404 Greg Childers      
2383 1710 7705 Greg Childers      
2385 8 2154 Andrey Kulsha Greg Childers 2001-August-11 Primo 0.1.0 (ECPP)
2408 975 7198 Peter Liaskovsky      
2432 891 7175 Peter Liaskovsky François Morain 2005-March-8 fastECPP
2438 1995 8046 Peter Liaskovsky François Morain 2003-September-3 ECPP-V11.0.5alpha
2448 535 6680 Peter Liaskovsky      
2453 2094 8147 Peter Liaskovsky      
2460 671 6954 Peter Liaskovsky      
2463 1274 7649 Peter Liaskovsky      
2470 1249 7649 Peter Liaskovsky      
2473 1188 7605 Peter Liaskovsky      
2481 2432 8401 Peter Liaskovsky      
2489 1858 8137 Peter Liaskovsky      
2494 635 6991 Peter Liaskovsky      
2495 2424 8445 Peter Liaskovsky      
2518 1647 8100 Peter Liaskovsky François Morain 2005-March-24 FastECPP
2522 537 6885 Peter Liaskovsky      
2528 2031 8362 Peter Liaskovsky      
2543 414 6656 Peter Liaskovsky      
2551 622 7127 Peter Liaskovsky François Morain 2003-July-16 ECPP-V11.0.5alpha
2553 974 7630 Peter Liaskovsky      
2573 1134 7860 Peter Liaskovsky      
2589 218 6055 Peter Liaskovsky François Morain 2003-July-16 ECPP-V11.0.5alpha
2596 1867 8492 Peter Liaskovsky      
2613 2348 8808 Paul Leyland      
2622 91 5137 Paul Leyland      
2622 2129 8727 Paul Leyland      
2625 1094 7978 Paul Leyland      
2659 2658 9106 Henri Lifchitz François Morain 2005-June-10 FastECPP
2665 1702 8611 Paul Leyland      
2668 199 6134 Paul Leyland      
2674 477 7163 Paul Leyland François Morain 2004-June-9 ECPP-V11.0.5alpha
2675 298 6619 Paul Leyland      
2680 2053 8878 Paul Leyland      
2680 2319 9020 Paul Leyland François Morain 2004-April-30 ECPP-V11.0.5alpha
2685 1904 8806 Paul Leyland      
2696 2451 9138 Paul Leyland      
2716 2197 9077 Greg Childers François Morain 2005-January-3 ECPP-V11.0.5alpha
2722 2445 9223 Greg Childers      
2759 2200 9222 Greg Childers      
2763 14 3167 Greg Childers François Morain 2003-September-5 ECPP-V11.0.5alpha
2771 2640 9482 Greg Childers      
2779 1632 8929 Greg Childers      
2779 2560 9472 Greg Childers      
2803 916 8303 Greg Childers      
2823 836 8250 Greg Childers      
2826 1289 8790 Greg Childers      
2831 666 7994 Greg Childers      
2843 208 6591 Greg Childers      
2873 76 5404 Greg Childers      
2883 1136 8809 Greg Childers      
2890 1671 9315 Greg Childers      
2892 2035 9569 Greg Childers      
2900 117 5998 Greg Childers      
2930 2739 10073 Greg Childers François Morain 2005-January-17 mpifastecpp
2974 2735 10222 Greg Childers      
2987 2680 10240 Greg Childers      
2996 1563 9570 Greg Childers      
3028 483 8127 Sander Hoogendoorn      
3042 2231 10187 Sander Hoogendoorn      
3070 1781 9980 Peter Liaskovsky      
3076 7 2600 Andrey Kulsha Greg Childers 2001-September-10 Primo 1.0 (ECPP)
3081 1568 9845 Peter Liaskovsky      
3099 1078 9399 Peter Liaskovsky      
3100 11 3229 Sander Hoogendoorn Jonathan A. Zylstra 2003-December-30 Primo v2.1.2 (ECPP)
3100 2447 10505 Peter Liaskovsky      
3107 712 8863 Alexander Kuzmich      
3110 63 5596 Andrey Kulsha      
3121 3070 10884 Alexander Kuzmich      
3142 3089 10966 Alexander Kuzmich      
3147 214 7334 Alexander Kuzmich      
3155 1966 10392 Alexander Kuzmich      
3160 2119 10511 Alexander Kuzmich      
3171 1256 9827 Alexander Kuzmich      
3178 1005 9541 Alexander Kuzmich      
3178 1543 10133 Alexander Kuzmich      
3185 1182 9787 Alexander Kuzmich      
3188 2781 10981 Alexander Kuzmich      
3201 1772 10399 Leonid Muraviov      
3213 2942 11145 Leonid Muraviov      
3217 2118 10700 Leonid Muraviov      
3222 553 8838 Leonid Muraviov      
3225 1678 10400 Leonid Muraviov      
3237 82 6196 Andrey Kulsha      
3238 211 7527 Leonid Muraviov      
3243 2482 11010 Leonid Muraviov      
3265 1234 10094 Leonid Muraviov François Morain 2005-June-10 FastECPP
3268 1755 10603 Leonid Muraviov      
3369 1766 10940 Leonid Muraviov      
3378 2669 11575 Leonid Muraviov      
3392 1209 10456 Andrey Kulsha      
3395 1966 11182 Andrey Kulsha      
3423 2560 11667 Andrey Kulsha      
3423 3070 11937 Andrey Kulsha      
3442 2197 11503 Andrey Kulsha      
3469 2248 11628 Andrey Kulsha      
3470 1173 10651 Andrey Kulsha      
3502 1271 10871 Paul Leyland      
3507 3172 12280 Paul Leyland      
3509 1330 10962 Paul Leyland      
3514 197 8063 Paul Leyland François Morain 2004-August-23 ECPP-V11.05alpha
3522 851 10320 Paul Leyland      
3543 1262 10988 Paul Leyland      
3543 3052 12346 Paul Leyland      
3557 1516 11314 Paul Leyland      
3571 648 10041 Paul Leyland François Morain 2003-December-18 ECPP-V11.0.5alpha
3594 697 10219 Paul Leyland      
3602 2577 12287 Paul Leyland      
3646 2389 12317 Paul Leyland      
3650 2613 12473 Paul Leyland      
3710 303 9207 Paul Leyland      
3712 507 10041 Paul Leyland      
3712 1407 11687 Paul Leyland      
3712 3429 13123 Paul Leyland      
3722 75 6979 Andrey Kulsha      
3729 2482 12660 Paul Leyland      
3759 2 1132 Rob Binnekamp Andrey Kulsha 2001-June-9 Titanix 2.1.0 (ECPP)
3805 922 11281 Paul Leyland      
3836 2445 12998 Paul Leyland      
3841 2580 13105 Paul Leyland      
3858 2095 12814 Paul Leyland      
3867 2572 13188 Paul Leyland      
3874 2985 13462 Paul Leyland      
3892 765 11224 Paul Leyland      
3894 2107 12943 Paul Leyland      
3896 579 10764 Paul Leyland      
3908 2109 12991 Paul Leyland      
3924 1325 12252 Paul Leyland      
3926 1689 12672 Paul Leyland      
3938 3321 13867 Paul Leyland      
3976 2607 13583 Paul Leyland      
3987 3128 13936 Paul Leyland      
3992 2361 13466 Paul Leyland      
3993 1778 12977 Paul Leyland      
3996 539 10916 Paul Leyland      
4015 1924 13187 Paul Leyland      
4023 1378 12630 Paul Leyland      
4058 3075 14154 Paul Leyland      
4071 2746 13999 Paul Leyland      
4130 2133 13749 Paul Leyland      
4153 2154 13843 Paul Leyland      
4162 1727 13474 Paul Leyland      
4163 3348 14674 Paul Leyland      
4180 3911 15016 Paul Leyland      
4189 496 11292 Paul Leyland      
4192 2513 14254 Paul Leyland      
4198 3521 14889 Paul Leyland      
4211 2772 14498 Paul Leyland      
4217 472 11277 Paul Leyland      
4241 2580 14469 Paul Leyland      
4263 3694 15209 Paul Leyland      
4266 1961 14046 Paul Leyland      
4278 1027 12884 Paul Leyland      
4307 3976 15503 Paul Leyland      
4309 298 10662 Paul Leyland      
4312 3739 15406 Paul Leyland      
4342 1991 14325 Paul Leyland      
4359 1916 14308 Paul Leyland      
4364 1335 13640 Paul Leyland      
4365 1724 14128 Paul Leyland      
4398 3563 15621 Paul Leyland      
4400 1347 13770 Paul Leyland      
4400 3657 15678 Paul Leyland      
4405 2638 15071 Greg Childers Franke, Kleinjung, Morain & Wirth 2004-July-20 fastECPP
4414 1093 13413 Greg Childers      
4444 535 12125 Rob Binnekamp      
4457 4134 16119 Greg Childers      
4468 61 7977 Andrey Kulsha      
4471 2940 15507 Greg Childers      
4476 1945 14722 Greg Childers      
4491 3856 16106 Greg Childers      
4498 2979 15627 Greg Childers      
4504 165 9988 Mark Rodenkirch      
4525 4104 16350 Mark Rodenkirch      
4534 4375 16509 Mark Rodenkirch      
4537 2248 15208 Mark Rodenkirch      
4554 1637 14637 Mark Rodenkirch      
4558 1715 14742 Mark Rodenkirch      
4568 3507 16194 Mark Rodenkirch      
4575 2546 15582 Mark Rodenkirch      
4600 3937 16538 Mark Rodenkirch      
4608 3407 16278 Mark Rodenkirch      
4613 1716 14921 Mark Rodenkirch      
4613 4572 16885 Mark Rodenkirch      
4632 3331 16317 Mark Rodenkirch      
4644 4385 16914 Mark Rodenkirch      
4660 1583 14910 Mark Rodenkirch      
4673 2118 15543 Mark Rodenkirch      
4682 669 13229 Mark Rodenkirch      
4708 2425 15936 Mark Rodenkirch      
4716 71 8731 Andrey Kulsha      
4744 1385 14904 Mark Rodenkirch      
4791 4790 17633 Henri Lifchitz      
4805 4378 17497 Mark Rodenkirch      
4824 2665 16526 Mark Rodenkirch      
4838 3885 17366 Mark Rodenkirch      
4840 1087 14696 Mark Rodenkirch      
4852 2595 16566 Mark Rodenkirch      
4865 3972 17510 Mark Rodenkirch      
4866 1159 14910 Mark Rodenkirch      
4872 4667 17876 Mark Rodenkirch      
4881 3062 17016 Mark Rodenkirch      
4890 3769 17488 Mark Rodenkirch      
4893 2294 16444 Mark Rodenkirch      
4897 4540 17909 Mark Rodenkirch      
4926 461 13122 Mark Rodenkirch      
4928 4365 17938 Mark Rodenkirch      
4945 4452 18043 Mark Rodenkirch      
4949 816 14410 Mark Rodenkirch      
4949 2406 16735 Mark Rodenkirch      
4949 4866 18248 Mark Rodenkirch      
4974 583 13757 Mark Rodenkirch      
4990 1687 16104 Mark Rodenkirch      
4991 3046 17388 Mark Rodenkirch      
5014 1577 16034 Paul Leyland      
5016 3485 17768 Paul Leyland      
5026 499 13561 Paul Leyland      
5031 1946 16548 Paul Leyland      
5041 70 9302 Andrey Kulsha      
5043 3356 17781 Paul Leyland      
5060 5019 18726 Paul Leyland      
5062 4611 18547 Paul Leyland      
5078 4413 18508 Paul Leyland      
5090 9 4858 Andrey Kulsha      
5099 4978 18852 Paul Leyland      
5128 801 14890 Paul Leyland      
5140 471 13740 Paul Leyland      
5154 2255 17283 Paul Leyland      
5155 3384 18195 Paul Leyland      
5182 1799 16868 Paul Leyland      
5234 3777 18723 Greg Childers      
5241 1682 16907 Greg Childers      
5245 2932 18186 Greg Childers      
5248 257 12648 Greg Childers      
5256 2317 17687 Greg Childers      
5266 1765 17098 Greg Childers      
5277 2200 17638 Greg Childers      
5327 3232 18695 Greg Childers      
5336 4323 19401 Greg Childers      
5338 2923 18501 Greg Childers      
5376 2419 18191 Greg Childers      
5385 2852 18606 Greg Childers      
5411 3958 19466 Greg Childers      
5421 2518 18438 Greg Childers      
5425 2544 18475 Greg Childers      
5471 900 16163 Greg Childers      
5488 99 10953 Andrey Kulsha      
5501 4842 20272 Anatoly Selevich      
5509 4938 20348 Anatoly Selevich      
5516 3735 19705 Anatoly Selevich      
5538 2749 19047 Anatoly Selevich      
5542 3947 19931 Anatoly Selevich      
5563 5092 20622 Anatoly Selevich      
5670 229 13381 Anatoly Selevich      
5678 3777 20312 Anatoly Selevich      
5685 224 13362 Anatoly Selevich      
5694 1433 17972 Anatoly Selevich      
5698 381 14707 Anatoly Selevich      
5713 2772 19669 Anatoly Selevich      
5738 885 16910 Anatoly Selevich      
5755 376 14821 Anatoly Selevich      
5757 3562 20448 Anatoly Selevich      
5763 4612 21115 Anatoly Selevich      
5769 4316 20971 Anatoly Selevich      
5776 2955 20046 Anatoly Selevich      
5776 4667 21193 Anatoly Selevich      
5792 921 17169 Anatoly Selevich      
5797 2674 19868 Anatoly Selevich      
5802 5033 21478 Anatoly Selevich      
5805 4508 21212 Anatoly Selevich      
5809 5286 21628 Anatoly Selevich      
5815 2544 19804 Anatoly Selevich      
5827 412 15238 Anatoly Selevich      
5841 2668 20013 Anatoly Selevich      
5845 4024 21070 Anatoly Selevich      
5852 159 12883 Anatoly Selevich      
5876 345 14913 Anatoly Selevich      
5892 1793 19171 Anatoly Selevich      
5920 553 16237 Anatoly Selevich      
5935 1792 19309 Anatoly Selevich      
5942 1335 18572 Anatoly Selevich      
5947 2434 20139 Anatoly Selevich      
5958 3889 21389 Anatoly Selevich      
5963 1234 18434 Anatoly Selevich      
5968 2593 20374 Anatoly Selevich      
5988 1133 18289 Anatoly Selevich      
6030 2471 20460 Anatoly Selevich      
6041 3148 21132 Anatoly Selevich      
6056 345 15370 Anatoly Selevich      
6058 465 16160 Anatoly Selevich      
6082 4119 21986 Anatoly Selevich      
6114 5083 22660 Anatoly Selevich      
6129 4532 22410 Anatoly Selevich      
6156 3193 21572 Anatoly Selevich      
6168 1759 20017 Anatoly Selevich      
6182 4953 22842 Anatoly Selevich      
6201 778 17927 Anatoly Selevich      
6219 3140 21748 Anatoly Selevich      
6255 1496 19860 Anatoly Selevich      
6258 3583 22243 Anatoly Selevich      
6274 313 15658 Anatoly Selevich      
6312 1427 19911 Anatoly Selevich      
6318 2789 21769 Anatoly Selevich      
6325 2628 21630 Anatoly Selevich      
6411 2492 21776 Anatoly Selevich      
6430 839 18800 Anatoly Selevich      
6436 5253 23945 Anatoly Selevich      
6436 6047 24338 Anatoly Selevich      
6438 5981 24315 Anatoly Selevich      
6454 3505 22878 Anatoly Selevich      
6471 1858 21154 Anatoly Selevich      
6471 3572 22991 Anatoly Selevich      
6482 195 14845 Anatoly Selevich      
6482 5247 24113 Anatoly Selevich      
6538 3445 23127 Anatoly Selevich      
6538 5869 24639 Anatoly Selevich      
6542 2583 22323 Anatoly Selevich      
6563 3378 23159 Anatoly Selevich      
6578 633 18428 Anatoly Selevich      
6582 5977 24857 Anatoly Selevich      
6586 1711 21295 Anatoly Selevich      
6605 4572 24176 Anatoly Selevich      
6607 2854 22831 Anatoly Selevich      
6612 485 17759 Anatoly Selevich      
6623 774 19133 Anatoly Selevich      
6628 1033 19978 Anatoly Selevich      
6628 1603 21243 Anatoly Selevich      
6639 272 16164 Anatoly Selevich      
6669 1312 20794 Anatoly Selevich      
6671 5062 24712 Anatoly Selevich      
6682 103 13450 Anatoly Selevich      
6706 3865 24056 Anatoly Selevich      
6707 6462 25557 Anatoly Selevich      
6738
3751 24083 Anatoly Selevich      
6739 3100 23529 Anatoly Selevich      
6753 5122 25050  Anatoly Selevich      
6772 713 19322  Anatoly Selevich      
6786 1009 20385  Anatoly Selevich      
6807 3434 24069  Anatoly Selevich      
6812 1917 22362  Anatoly Selevich      
6814 545 18646 Anatoly Selevich      
6814 3053 23745 Anatoly Selevich      
6825 1496 21669 Anatoly Selevich      
6861 3410 24239 Anatoly Selevich      
6869 5956 25931 Anatoly Selevich      
6886 29 10071 Andrey Kulsha      
6891 5830 25950 Anatoly Selevich      
6897 1280 21431 Anatoly Selevich      
6918 91 13553 Andrey Kulsha      
6935 2704 23801 Anatoly Selevich      
6965 6694 26646 Anatoly Selevich      
6967 516 18900 Anatoly Selevich      
6979 5680 26202 Anatoly Selevich      
6994 1673 22546 Anatoly Selevich      
6995 1926 22977 Anatoly Selevich      
7002 6961 26907 Anatoly Selevich      
7017 3590 24947 Anatoly Selevich      
7022 1095 21343 Anatoly Selevich      
7030 4167 25448 Anatoly Selevich      
7035 34 10774 Andrey Kulsha      
7044 415 18442 Anatoly Selevich      
7048 2947 24453 Anatoly Selevich      
7060 2399 23864 Anatoly Selevich      
7070 3729 25252 Anatoly Selevich      
7074 5227 26303 Anatoly Selevich      
7100 4953 26234 Anatoly Selevich      
7115 1242 22015 Anatoly Selevich      
7121 2200 23802 Anatoly Selevich      
7123 7116 27440 Anatoly Selevich      
7130 2841 24624 Anatoly Selevich      
7132 3027 24827 Anatoly Selevich      
7155 154 15652 Anatoly Selevich      
7157 3436 25308 Anatoly Selevich      
7166 9 6839 Andrey Kulsha      
7166 6045 27098 Anatoly Selevich      
7172 513 19437 Anatoly Selevich      
7179 6512 27379 Anatoly Selevich      
7193 1936 23643 Anatoly Selevich      
7205 1702 23280 Anatoly Selevich      
7258 4343 26404 Anatoly Selevich      
7283 2680 24968 Anatoly Selevich      
7288 3489 25820 Anatoly Selevich      
7300 91 14302 Andrey Kulsha      
7312 371 18788 Anatoly Selevich      
7312 5269 27214 Anatoly Selevich      
7316 4911 27005 Anatoly Selevich      
7321 5310 27272 Anatoly Selevich      
7334 6795 28106 Anatoly Selevich      
7372 107 14961 Anatoly Selevich      
7394 3933 26580 Anatoly Selevich      
7394 5907 27886 Anatoly Selevich      
7414 6783 28407 Anatoly Selevich      
7429 3628 26445 Anatoly Selevich      
7637 916 22620 Raffi Chaglassian      
7438 5311 27708 Anatoly Selevich      
7439 5388 27759 Anatoly Selevich      
7448 6445 28470 Anatoly Selevich      
7452 6583 28455 Anatoly Selevich      
7453 5388 27811 Anatoly Selevich      
7468 7161 28789 Anatoly Selevich      
7469 2008 24669 Anatoly Selevich      
7474 6247 28369 Anatoly Selevich      
7507 4834 27659 Anatoly Selevich      
7539 2270 25302 Anatoly Selevich      
7539 3596 26808 Anatoly Selevich      
7555 142 16261 Raffi Chaglassian      
7574 2607 25874 Anatoly Selevich      
7579 2092 25167 Anatoly Selevich      
7581 6812 29061 Anatoly Selevich      
7592 4935 28040 Anatoly Selevich      
7596 3281 26708 Anatoly Selevich      
7601 1468 24071 Anatoly Selevich      
7614 4547 27850 Anatoly Selevich      
7618 831 22242 Raffi Chaglassian      
7619 360 19477 Raffi Chaglassian      
7621 2982 26480 Anatoly Selevich      
7660 1347 23971 Anatoly Selevich      
7696 753 22140 Raffi Chaglassian      
7711 1132 23549 Anatoly Selevich      
7761 1072 23518 Anatoly Selevich      
7873 870 23143 Raffi Chaglassian      
7888 863 23160 Raffi Chaglassian      
7949 898 23476 Raffi Chaglassian      
8025 1142 24538 Anatoly Selevich      
8048 1383 25278 Anatoly Selevich      
8062 85 15555 Andrey Kulsha      
8117 580 22431 Anatoly Selevich      
8151 578 22513 Anatoly Selevich      
8177 1792 26603 Anatoly Selevich      
8200 249 19649 Anatoly Selevich      
8217 1142 25125 Anatoly Selevich      
8266 1631 26555 Anatoly Selevich      
8332 1311 25976 Anatoly Selevich      
8340 193 19062 Raffi Chaglassian      
8604 415 22526 Anatoly Selevich      
8606 579 23776 Anatoly Selevich      
8655 938 25725 Anatoly Selevich      
8685 518 23574 Anatoly Selevich      
8812 847 25801 Anatoly Selevich      
8907 5372 33225 Anatoly Selevich      
8925 4514 32617 Anatoly Selevich      
8942 4785 32906 Anatoly Selevich      
8946 5855 33705 Anatoly Selevich      
8961 4342 32598 Anatoly Selevich      
9112 921 27011 Anatoly Selevich      
9357 202 21572 Anatoly Selevich      
9363 392 24281 Anatoly Selevich      
9377 9376 37246 Henri Lifchitz      
9406 711 26825 Anatoly Selevich      
9725 988 29125 Anatoly Selevich      
9744 865 28619 Anatoly Selevich      
9820 129 20726 Raffi Chaglassian      
9892 257 23840 Anatoly Selevich      
9923 580 27422 Anatoly Selevich      
10029 572 27654 Anatoly Selevich      
10112 3 4825 Andrey Kulsha Andrey Kulsha 2004-February-7 Primo 2.0.0b3 (ECPP)
10188 749 29286 Anatoly Selevich      
10392 665 29335 Anatoly Selevich      
10417 532 28396 Anatoly Selevich      
10421 730 29839 Anatoly Selevich      
10479 470 28001 Anatoly Selevich      
10747 54 18619 Alexander Kuzmich      
10960 41 17677 Alexander Kuzmich      
11006 99 21964 Alexander Kuzmich      
11290 607 31423 Anatoly Selevich      
11554 747 33199 Anatoly Selevich      
11613 422 30488 Anatoly Selevich      
11666 135 24853 Donovan Johnson      
11796 7 9969 Andrey Kulsha      
11889 8 10737 Andrey Kulsha      
11962 103 24078 Donovan Johnson      
11974 817 34871 Anatoly Selevich      
12130 521 32956 Anatoly Selevich      
12159 836 35532 Anatoly Selevich      
12483 202 28778 Anatoly Selevich      
12503 514 33896 Anatoly Selevich      
12698 513 34414 Anatoly Selevich      
12717 236 30177 Anatoly Selevich      
12883 160 28396 Donovan Johnson      
12903 488 34689 Anatoly Selevich      
13130 189 29890 Donovan Johnson      
13442 279 32874 Anatoly Selevich      
13620 829 39751 Anatoly Selevich      
13715 694 38970 Anatoly Selevich      
13934 117 28819 Donovan Johnson      
14415 604 40089 Anatoly Selevich      
14494 933 43046 Anatoly Selevich      
14523 574 40068 Anatoly Selevich      
14577 368 37403 Anatoly Selevich      
14811 56 25893 Andrey Kulsha      
14931 712 42591 Anatoly Selevich      
15044 15 17694 Andrey Kulsha      
15178 137 32432 Andrey Kulsha      
15601 286 38322 Anatoly Selevich      
16725 638 46911 Anatoly Selevich      
17200 713 49074 Anatoly Selevich      
17203 376 44301 Anatoly Selevich      
17799 322 44638 Anatoly Selevich      
17873 118 37031 Andrey Kulsha      
17994 955 53623 Anatoly Selevich      
18078 349 45970 Anatoly Selevich      
18225 622 50917 Anatoly Selevich      
18348 425 48226 Anatoly Selevich      
18388 471 49152 Anatoly Selevich      
18514 95 36616 Andrey Kulsha      
19025 184 43089 Andrey Kulsha      
19312 429 50839 Anatoly Selevich      
19317 788 55953 Anatoly Selevich      
19642 169 43761 Andrey Kulsha      
19674 53 33924 Andrey Kulsha      
19783 492 53256 Anatoly Selevich      
19846 455 52751 Anatoly Selevich      
26205 8 23666 Peter Liaskovsky      
29355 2 8837 Rob Binnekamp      
34653 2 10432 Rob Binnekamp      
57285 2 17245 Rob Binnekamp      
58046 9 55390 Andrey Kulsha      
63880 3 30479 Andrey Kulsha      
78296 3 37357 Andrey Kulsha      
91382 9 87201 Andrey Kulsha      
99069 2 29823 Rob Binnekamp