abc Conjecture - Computational Results

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.

Distribution of triples with q > 1
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.

Highest quality triples
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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