Computational Results
In 2006, the Mathematics Department of Leiden University in the Netherlands, together with the Dutch Kennislink science institute, launched the ABC@Home project, a grid computing system which aims to discover additional triples a, b, c with rad(abc) < c. Although no finite set of examples or counterexamples can resolve the abc conjecture, it is hoped that patterns in the triples discovered by this project will lead to insights about the conjecture and about number theory more generally.
| q > 1 | q > 1.05 | q > 1.1 | q > 1.2 | q > 1.3 | q > 1.4 | |
|---|---|---|---|---|---|---|
| c < 102 | 6 | 4 | 4 | 2 | 0 | 0 |
| c < 103 | 31 | 17 | 14 | 8 | 3 | 1 |
| c < 104 | 120 | 74 | 50 | 22 | 8 | 3 |
| c < 105 | 418 | 240 | 152 | 51 | 13 | 6 |
| c < 106 | 1,268 | 667 | 379 | 102 | 29 | 11 |
| c < 107 | 3,499 | 1,669 | 856 | 210 | 60 | 17 |
| c < 108 | 8,987 | 3,869 | 1,801 | 384 | 98 | 25 |
| c < 109 | 22,316 | 8,742 | 3,693 | 706 | 144 | 34 |
| c < 1010 | 51,677 | 18,233 | 7,035 | 1,159 | 218 | 51 |
| c < 1011 | 116,978 | 37,612 | 13,266 | 1,947 | 327 | 64 |
| c < 1012 | 252,856 | 73,714 | 23,773 | 3,028 | 455 | 74 |
| c < 1013 | 528,275 | 139,762 | 41,438 | 4,519 | 599 | 84 |
| c < 1014 | 1,075,319 | 258,168 | 70,047 | 6,665 | 769 | 98 |
| c < 1015 | 2,131,671 | 463,446 | 115,041 | 9,497 | 998 | 112 |
| c < 1016 | 4,119,410 | 812,499 | 184,727 | 13,118 | 1,232 | 126 |
| c < 1017 | 7,801,334 | 1,396,909 | 290,965 | 17,890 | 1,530 | 143 |
| c < 1018 | 14,482,065 | 2,352,105 | 449,194 | 24,013 | 1,843 | 160 |
As of September 2012, ABC@Home has found 23.1 million triples, and its present goal is to obtain a complete list of all ABC triples (a,b,c) with c no more than 1020.
| q | a | b | c | Discovered by | |
|---|---|---|---|---|---|
| 1 | 1.6299 | 2 | 310·109 | 235 | Eric Reyssat |
| 2 | 1.6260 | 112 | 32·56·73 | 221·23 | Benne de Weger |
| 3 | 1.6235 | 19·1307 | 7·292·318 | 28·322·54 | Jerzy Browkin, Juliusz Brzezinski |
| 4 | 1.5808 | 283 | 511·132 | 28·38·173 | Jerzy Browkin, Juliusz Brzezinski, Abderrahmane Nitaj |
| 5 | 1.5679 | 1 | 2·37 | 54·7 | Benne de Weger |
Note: the quality q(a, b, c) of the triple (a, b, c) is defined above.
Read more about this topic: abc Conjecture
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