### Triples of small size

At the Mathematical Institute in Leiden there have been several activities around the ABC conjecture and enumerating ABC-triples of small size, starting with a meeting in September 2005 of the Intercity Number Theory Seminar. In 2007 we started a science promotion project in the Netherlands called Reken mee met ABC, which is also the subject of a paper paper in Dutch [PDF, March 2007] that I wrote with Gillien Geuze.

### ABC@Home

One part of the project was a volunteer network called ABC@home [wikipedia] devoted to a distributed computational project that would come up with a complete list of all ABC triples consisting of integers of at most 18 digits. We reached this goal in 2011 with a list of 14482065 triples. The triples are available in the file abctriples_below_1018.gz (gzipped text file, 179 MB). For each triple (a, b, c) with c below 1018 the file has a line with the numbers c and a.

See chapter 7 of the PhD thesis of WIllem Jan Palenstijn for a description of the algorithm used and why we believe that this list is complete. The chapter also contains some interesting observations about the list of triples.

The bachelor thesis of Thijs van Dijk from 2011 addresses the implementation.

In the years 2012-2015 ABC@home has been finding additional triples (a,b,c) with with c in the range 1018 ≤ c < 263. This produced 9345651 additional triples that are available in the file big_triples.gz (gzipped text file, 133 MB).

At this moment ABC@home is no longer active.

### Table

The list of all 418 ABC triples consisting of numbers of at most 5 digits, sorted by size, is given below. The file abctriples_below_1018.gz contains the smallest 14 million ABC triples.

qualitysizemerita bc
11.22630.950.0512332
21.01901.510.0053325
31.04121.690.01124​372
41.11271.810.06132​726
51.29201.910.36124​534
61.17571.910.15257234
71.02722.100.012211253
81.42662.110.7635327
91.01292.350.00125​732​52
101.31112.390.5812·​11235
111.02882.390.012355·​72
121.03262.400.017352·​53
131.27282.410.48133528
141.03702.410.013452​728
151.22522.460.36125​32172
161.09182.540.0822​523573
171.10842.570.1125733·​53
181.04562.710.0253·​13229
191.19882.710.341327329
201.31762.710.7412933​19
211.02512.730.01332972​11
221.07902.800.07124​3·​1354
231.20402.800.377226​3254
241.03262.800.013425​1754
251.09222.830.09133​5222​132
261.04592.860.02123​7·​1336
271.13672.860.195226​1136
281.10482.860.1223​135436
291.00842.860.0023​5223236
301.00482.980.00126​3·​5312
311.03442.990.01735423​112
321.15233.010.25121052​41
331.29723.010.7952103·​73
341.11943.080.17135​526​19
351.17253.120.332333​72113
361.24053.120.59342·​54113
371.02433.120.013526​17113
381.02993.140.0127​536372
391.00613.200.00281133·​232
401.04393.230.033426​52412
411.16643.310.342334​52211
421.07673.310.08527·​172211
431.20163.310.48355·​192211
441.08633.310.1032211112​17
451.09503.340.121127​1737
461.14333.340.2613921137
471.01113.340.002952​6737
481.28983.340.892·​537133
491.02703.340.013422​232133
501.02043.360.01172​4728​32
511.24243.360.67533723​172
521.45573.381.86125​3·​5274
531.00503.380.005223​33​1174
541.02583.380.0126​347274
551.18463.380.4221034​1774
561.01023.400.0032​117422​54
571.36073.431.335421135​11
581.03483.480.02124​33​752​112
591.01493.490.0053​7133210​3
601.09173.490.1253210​355
611.14763.490.302734​3755
621.21133.500.56115526​72
631.10593.510.1621037132​19
641.09043.540.125227​33592
651.00053.550.0033​175529​7
661.06333.580.062736​573​11
671.06333.580.0623​345573​11
681.10323.590.161132​2324​35
691.15543.600.34127​3134​72
701.00303.600.00295923·​113
711.07093.600.0873·​11325​53
721.05133.610.04132​5·​7·​13212
731.08803.610.125311·​192212
741.10733.610.173·​53612212
751.06663.610.07112123·​372
761.01783.620.015·​721235​17
771.05523.630.052834​7252​132
781.07833.630.10211375·​7·​112
791.56793.642.8912·​3754​7
801.00993.690.007228​19173
811.09453.700.1427173712
821.04413.710.031736​7210​5
831.09693.730.15375526​83
841.02443.730.0111321234​67
851.11513.750.21732​5429​11
861.04463.760.037433​5324​192
871.31963.771.26173​1723​36
881.05213.790.051953​72211​3
891.12483.800.25327922·​55
901.12783.810.26134​7928​52
911.23533.820.77125​5·​4138
921.03613.820.027·​2328​5238
931.33763.821.4117227​7238
941.13103.820.2726​579238
951.00093.820.0031225​52​738
961.03833.820.0321072​11338
971.03703.820.0247228​1738
981.11083.820.207426​5·​1338
991.19363.820.55263853​53
1001.05253.820.05263·​1335·​113
1011.21173.820.6515·​11329​13
1021.06533.840.0724​7235​52193
1031.22813.840.74119322​5·​73
1041.15583.840.382419354​11
1051.01553.840.005319328​33
1061.02923.890.027·​1323826​112
1071.09863.900.1732​7317326​53
1081.02813.900.021926​5336​11
1091.01753.910.013272​167213
1101.02343.910.011134​101213
1111.07143.930.09152​7327​67
1121.00183.940.00735·​41222​37
1131.00283.970.00245·​43233​73
1141.18013.970.5132​532137·​113
1151.15593.970.407·​1322133·​55
1161.00713.970.0013·​5525​293
1171.10073.970.18126​3·​72972
1181.18663.990.55123​52​7234​112
1191.06493.990.085234​1122·​173
1201.03354.000.025519328​3·​13
1211.04674.000.0413333​17224​54
1221.06584.030.08132​7·​13223​113
1231.20074.040.6411322​7437​5
1241.02684.040.014137​525​73
1251.17604.040.513225​735·​133
1261.00194.040.002128325·​133
1271.10514.070.20743·​5529​23
1281.13194.080.302554​1935​72
1291.07404.080.10210​519332​113
1301.02154.080.01335·​7428​47
1311.02154.080.015335​7228​47
1321.00394.090.00172·​35​52233
1331.10894.090.222·​345·​74233
1341.25554.090.99123323​32​132
1351.28554.091.19112233212​3
1361.01174.090.0029109236​17
1371.11694.110.257427​341132
1381.22464.160.811211​735​59
1391.14684.170.382432​53​13114
1401.05634.170.067228​3·​19114
1411.03464.170.0223225​32​72114
1421.00234.170.0072​312·​38114
1431.00834.170.0041225​34​5114
1441.08154.170.1317329​19114
1451.07024.170.102·​721143·​173
1461.15754.170.443511422​612
1471.01174.180.002253​11232​412
1481.03424.180.0253222​5537​7
1491.09974.190.19123​32​7·​3156
1501.28554.191.233228​6156
1511.01014.190.00243·​112​4356
1521.05214.190.06192·​33​17256
1531.25554.191.027326​3556
1541.00384.190.0013727​11256
1551.04994.190.052·​347·​47256
1561.16464.190.483624​72​1956
1571.01514.190.0023​3·​4111456
1581.02624.190.0137224​34​1156
1591.02464.190.0135​722​59256
1601.15454.190.4224​11234​13256
1611.02104.190.0121233​7·​6156
1621.00494.190.0071223​33​7256
1631.06104.190.0713·​23222​3756
1641.11464.190.25235634​193
1651.12894.210.31128​32​71272
1661.17254.210.523253​131214
1671.25294.211.017233​5·​112214
1681.08614.210.153·​11·​2356214
1691.06544.210.0952​4337​7214
1701.05514.210.0647234​52​7214
1711.06624.210.0935​1352​232214
1721.08664.210.1532214132​97
1731.01674.220.0124​5311432​432
1741.14674.220.3921352​7336​23
1751.14704.230.3923​536​2375
1761.28234.231.2332​4721475
1771.07154.230.102·​3·​19211475
1781.11834.230.2636​1122​13375
1791.07384.230.113752·​5·​412
1801.18994.230.6222​177533​54
1811.01424.230.0049121433​54
1821.19144.230.631127525​232
1831.17814.230.5529​3561312
1841.04214.240.045533​232210​17
1851.05994.240.0773223323​37
1861.12874.240.31367527​137
1871.00394.240.00123​13334​7·​31
1881.02094.260.017228​7136​52
1891.26674.261.1329​711436​52
1901.11494.260.25173210​1336​52
1911.00344.260.003·​5421419·​312
1921.24144.270.9553​1375211​32
1931.01594.270.013721472​379
1941.02014.270.01742145·​13·​172
1951.00984.290.003454​31210​19
1961.06124.290.0822​57·​53239
1971.22694.290.873122​17339
1981.01244.290.008324​52​7239
1991.19254.290.6552​1923​7439
2001.04064.290.04293928​7·​11
2011.01584.300.01263972​13·​31
2021.03174.300.0211222​17332​133
2031.00664.300.00533925​619
2041.08124.300.133173925​54
2051.05334.310.063573​59212​5
2061.02404.310.0127112​1323·​193
2071.03664.320.0317228​3452​292
2081.06704.330.092453​13236​29
2091.07844.330.136727524​113
2101.05764.340.072332​7427​132
2111.26304.341.133527​13255​7
2121.07034.340.10134​27126​73
2131.09264.380.182552​31237​11
2141.01034.380.00737​1129​47
2151.04654.390.05111·​47222​35​52
2161.17114.390.54115·​173213​3
2171.15324.400.457436​3123​55
2181.19814.410.71126​34​572​232
2191.07144.420.1123​2923974​11
2201.16204.450.502133953​223
2211.00414.450.002829·​31232​55
2221.13064.450.34132​552·​73​41
2231.17924.460.607225​34​11134
2241.15144.460.4532​5322​193134
2251.22674.460.913824​53​11134
2261.12734.460.3351342·​33​232
2271.04654.460.052413417·​412
2281.00374.460.002513434​353
2291.03534.460.033·​37134212​7
2301.02024.470.01154​4726​33​17
2311.00694.470.002·​8325635​112
2321.01844.470.0122​7·​19239313
2331.14244.470.40131325​72​19
2341.15924.480.492·​38751732
2351.10784.480.2428​537535​53
2361.12314.490.313829​4754​72
2371.00304.490.0033​1354​7228​112
2381.04094.490.0417·​2927527​35
2391.13274.490.36122​33​17274​13
2401.08214.510.1547231328​53
2411.32664.521.7271812215
2421.10614.520.245334​13·​31215
2431.03464.520.0332​72​19293215
2441.04064.520.04121532​11·​331
2451.48294.523.183721538​5
2461.00894.520.0056113​1325​3·​73
2471.24494.521.073·​13221552​113
2481.10204.530.225332​6122·​75
2491.06734.530.1012·​7534​5·​83
2501.02424.540.012536​475·​193
2511.10194.540.222455​117·​173
2521.00244.540.002736​477·​173
2531.04924.540.063·​5421573​101
2541.03004.540.0223112​17224​37
2551.10844.550.2561228​5336​72
2561.00774.550.005627​15736​72
2571.04864.560.055223​67233​113
2581.10674.560.243831329​71
2591.00754.570.00192​23134212​32
2601.06474.570.0911212​3254​59
2611.04444.610.05210199255​13
2621.00684.610.0034​9721555​13
2631.03134.620.027·​11255​1329​34
2641.11124.620.273129​3473​112
2651.01274.620.0011425​29237​19
2661.10674.630.253523​73253​73
2671.17734.640.631927​733·​114
2681.06864.640.1121154​673·​114
2691.15054.650.47107221532​173
2701.10344.650.2428​3429353​192
2711.00874.660.0021434​19254​73
2721.00424.660.0025​19255​1138​7
2731.06284.660.091932·​3952​432
2741.06084.670.09173133​1926​36
2751.00554.690.001113​3725​34​19
2761.00564.700.003623​53​722232
2771.00644.700.00132​52​223210​72
2781.27624.701.3853210​7237​23
2791.03984.700.0421113·​6123·​75
2801.07524.700.13126​7·​11334​54
2811.00724.700.0027​7223234​54
2821.20934.700.8622​734​54373
2831.01094.700.0023​293·​75373
2841.04374.700.053·​542·​293373
2851.15124.700.48211​1132​55373
2861.00304.710.00392·​56312​53
2871.01064.720.0011337324​32​192
2881.05424.720.075·​13373​11223​38
2891.05484.720.0713334​542·​74​11
2901.01264.730.0072​1323​3855​17
2911.01264.730.0024​1323·​7555​17
2921.03024.730.02210​73·​5611·​173
2931.10454.740.2521411·​59237​52
2941.00534.740.0028​337·​19352​133
2951.05884.740.081211​33112​457
2961.01344.740.005522922·​34​73
2971.07304.740.131733732·​34​73
2981.00044.750.003953​17229​109
2991.25404.761.22123922·​134
3001.13474.770.40135​24122​114
3011.32354.771.84123​112​61310
3021.01424.770.01225·​72​241310
3031.16574.770.587223​53​59310
3041.20704.770.8613229​5·​23310
3051.05904.770.085·​9722​114310
3061.13564.770.4023​1122412310
3071.06564.770.1024​5473​11·​13310
3081.05404.770.0722​59253​192310
3091.00064.770.0027​11273​127310
3101.03504.770.035625​23·​59310
3111.04884.770.0613102·​52​1181
3121.06354.770.107·​1731025​432
3131.26274.771.302731017·​592
3141.16454.770.5721274​2333​133
3151.01244.770.0023​733​13355​19
3161.54714.774.1773310211​29
3171.16244.770.56135​5·​7224​612
3181.01354.780.002235​13·​1952​74
3191.06724.780.1111229​32​1352​74
3201.16334.780.5724​6131052​74
3211.01584.780.0123​35241252​74
3221.04094.780.043974​1722​53​112
3231.01494.790.0123​5·​72310132​192
3241.06964.790.121143·​5622​7·​133
3251.04554.790.057229​11232​832
3261.03244.790.0321152​7433​112​19
3271.04674.790.055373​18128​35
3281.01354.800.007235​25722​56
3291.01644.800.01123​32​53​72512
3301.23864.811.113229​53112​232
3311.01714.810.015628​33​7112​232
3321.01834.810.0153​2329​11233​74
3331.00044.810.00129​12732​52​172
3341.28674.821.53313·​712216
3351.06304.820.10734​809216
3361.00584.820.002373​191216
3371.11444.820.3013235​269216
3381.02734.820.0231772​113216
3391.07664.820.1470933​74216
3401.04934.820.0613·​499310216
3411.14694.820.4732​29273​132216
3421.02304.820.0134​13152​133216
3431.12774.820.37892163·​55​7
3441.07904.820.1533​192162572
3451.01854.820.0123​53​73102572
3461.10884.830.2721133​7454​107
3471.07394.830.13543·​149222​75
3481.07704.840.14134​7·​112210​67
3491.06284.840.1025​5234​292413
3501.02064.840.017229​33​52632
3511.12204.840.34374​29212​17
3521.01784.840.0119354​10125​37
3531.01604.840.01103255​1925​37
3541.09364.850.21252​53235​172
3551.25164.861.2412·​33​11355​23
3561.12444.860.35210​133102692
3571.01354.860.00352·​19125·​114
3581.00354.870.00562·​35​11274​31
3591.05094.880.0719229​3·​7254​112
3601.00744.880.0027​35211254​112
3611.01794.880.01128​13·​2337​5·​7
3621.02354.890.0117212​1934​312
3631.01524.890.0122​7134​31257
3641.02854.890.027·​43212​1957
3651.00154.890.003·​1322·​197257
3661.06244.890.1029222​139257
3671.08544.890.1834​29211​3757
3681.02964.890.0233​7328​26957
3691.00614.890.0024​34​1113·​17357
3701.12624.890.372·​11436​6757
3711.05304.890.073132·​11·​13357
3721.22024.890.99212​32113​3157
3731.11834.890.3373​1132·​3957
3741.19124.890.7824573·​7·​612
3751.03564.890.03112572·​35​7·​23
3761.00584.890.00355725​31·​79
3771.14924.900.506075722​39
3781.06644.900.115424​1733·​74​11
3791.06494.900.1152​3122​39433
3801.08054.900.1621234​72​19433
3811.04554.900.053473​23327​54
3821.03874.900.0417336​10327​54
3831.05924.910.097422​3913·​792
3841.03064.910.0214923102·​55​13
3851.14164.910.4636113​61214​5
3861.10134.920.2517·​172​41210​34
3871.18034.920.71126​32​5·​29174
3881.09694.920.2322​4335​73174
3891.03194.920.03577210​34174
3901.10194.920.2527​3272​412174
3911.05474.920.087425​3·​5·​132174
3921.00154.920.0072​23228​32​52174
3931.21704.920.9821333​5325·​75
3941.01304.930.0074214​535​347
3951.16964.930.633824​1737·​233
3961.04344.930.053·​57·​23326​113
3971.43504.933.10210​75738​13
3981.01404.930.012·​5611·​17338​13
3991.04784.930.062938​135·​1312
4001.03024.930.0222​5435​732932
4011.02324.940.015·​1735​19228​73
4021.02164.940.013631·​53228​73
4031.05934.940.0953​173·​13428​73
4041.19784.940.841143·​29328​73
4051.13394.940.423954​10928​73
4061.08784.950.1941229​13236​112
4071.01704.950.0121452​132​1736​112
4081.05024.950.07367·​1132213​11
4091.18994.950.783·​133174213​11
4101.05244.960.07129​3·​5955​29
4111.12284.960.3621513·​67236​53
4121.11014.960.291136​53210​89
4131.02874.970.025·​727​36173​19
4141.01944.970.01213293237​43
4151.03934.980.04755722​34​293
4161.01864.990.011372​7124​35​52
4171.15414.990.547523​101239​5
4181.05225.000.073673​17224​792