Editing Abc conjecture (section)

==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.

{| class="wikitable" style="text-align:right;"
|+ Distribution of triples with ''q'' > 1<ref name="Ref_d">{{Citation|url=http://www.rekenmeemetabc.nl/?item=h_stats |archive-url=https://web.archive.org/web/20081222221716/http://rekenmeemetabc.nl/?item=h_stats |url-status=dead |archive-date=December 22, 2008 |title=Synthese resultaten |work=RekenMeeMetABC.nl |access-date=October 3, 2012 |language=nl }}.</ref>
|-
! scope="col" {{diagonal split header|''c''|''q''}}
! scope="col" | ''q'' > 1
! scope="col" | ''q'' > 1.05
! scope="col" | ''q'' > 1.1
! scope="col" | ''q'' > 1.2
! scope="col" | ''q'' > 1.3
! scope="col" | ''q'' > 1.4
|-
! scope="row" | ''c'' < 10<sup>2</sup>
| 6 || 4 || 4 || 2 || 0 || 0
|-
! scope="row" | ''c'' < 10<sup>3</sup>
| 31 || 17 || 14 || 8 || 3 || 1
|-
! scope="row" | ''c'' < 10<sup>4</sup>
| 120 || 74 || 50 || 22 || 8 || 3
|-
! scope="row" | ''c'' < 10<sup>5</sup>
| 418 || 240 || 152 || 51 || 13 || 6
|-
! scope="row" | ''c'' < 10<sup>6</sup>
| 1,268 || 667 || 379 || 102 || 29 || 11
|-
! scope="row" | ''c'' < 10<sup>7</sup>
| 3,499 || 1,669 || 856 || 210 || 60 || 17
|-
! scope="row" | ''c'' < 10<sup>8</sup>
| 8,987 || 3,869 || 1,801 || 384 || 98 || 25
|-
! scope="row" | ''c'' < 10<sup>9</sup>
| 22,316 || 8,742 || 3,693 || 706 || 144 || 34
|-
! scope="row" | ''c'' < 10<sup>10</sup>
| 51,677 || 18,233 || 7,035 || 1,159 || 218 || 51					
|-
! scope="row" | ''c'' < 10<sup>11</sup>
| 116,978 || 37,612 || 13,266 || 1,947 || 327 || 64					
|-
! scope="row" | ''c'' < 10<sup>12</sup>
| 252,856 || 73,714 || 23,773 || 3,028 || 455 || 74					
|-
! scope="row" | ''c'' < 10<sup>13</sup>
| 528,275 || 139,762 || 41,438 || 4,519 || 599 || 84					
|-
! scope="row" | ''c'' < 10<sup>14</sup>
| 1,075,319 || 258,168 || 70,047 || 6,665 || 769 || 98					
|-
! scope="row" | ''c'' < 10<sup>15</sup>
| 2,131,671 || 463,446 || 115,041 || 9,497 || 998 || 112					
|-
! scope="row" | ''c'' < 10<sup>16</sup>
| 4,119,410 || 812,499 || 184,727 || 13,118 || 1,232 || 126					
|-
! scope="row" | ''c'' < 10<sup>17</sup>
| 7,801,334 || 1,396,909 || 290,965 || 17,890 || 1,530 || 143
|-
! scope="row" | ''c'' < 10<sup>18</sup>
| 14,482,065 || 2,352,105 || 449,194 || 24,013 || 1,843 || 160
|-
|}

As of May 2014, [[ABC@Home]] had found 23.8 million triples.<ref name="Ref_c">{{Citation |url=http://abcathome.com/data/ |title=Data collected sofar |work=ABC@Home |access-date=April 30, 2014 |url-status=dead |archive-url=https://web.archive.org/web/20140515021303/http://abcathome.com/data/ |archive-date=May 15, 2014 }}</ref>

{| class="wikitable"
|+ {{visible anchor|Highest-quality triples}}<ref>{{cite web |url=http://www.math.leidenuniv.nl/~desmit/abc/index.php?set=2 |title=100 unbeaten triples |work=Reken mee met ABC |date=2010-11-07 }}</ref>
|-
! scope="col" | Rank
! scope="col" | ''q''
! scope="col" | ''a''
! scope="col" | ''b''
! scope="col" | ''c''
! scope="col" class="unsortable" | Discovered by
|-
! scope="row" | 1
| 1.6299 || 2 || 3<sup>10</sup>·109 || 23<sup>5</sup> || Eric Reyssat
|-
! scope="row" | 2
| 1.6260 || 11<sup>2</sup> || 3<sup>2</sup>·5<sup>6</sup>·7<sup>3</sup> || 2<sup>21</sup>·23 || Benne de Weger
|-
! scope="row" | 3
| 1.6235 || 19·1307 || 7·29<sup>2</sup>·31<sup>8</sup> || 2<sup>8</sup>·3<sup>22</sup>·5<sup>4</sup> || Jerzy Browkin, Juliusz Brzezinski
|-
! scope="row" | 4
| 1.5808 || 283 || 5<sup>11</sup>·13<sup>2</sup> || 2<sup>8</sup>·3<sup>8</sup>·17<sup>3</sup> || Jerzy Browkin, Juliusz Brzezinski, Abderrahmane Nitaj
|-
! scope="row" | 5
| 1.5679 || 1 || 2·3<sup>7</sup> || 5<sup>4</sup>·7 || Benne de Weger
|}

Note: the ''quality'' ''q''(''a'', ''b'', ''c'') of the triple (''a'', ''b'', ''c'') is defined [[#Formulations|above]].