Numerical solutions to the (6,1,7) Diophantine equation from Peter J Ansell.
The integer equation where a sixth power equals the sum of seven sixth powers is known as the (6, 1, 7) equation.
Last update: 04 July 2009
Total number of solutions: 187
Finding primitive solutions to the (6, 1, 7) equation is a challenging exercise in analysis, programming and patience, as is the selection of the method used to present those solutions.
My approach is to split the equation
P^6 = Q0^6 + Q1^6 + Q2^6 + Q3^6 + Q4^6 + Q5^6 + Q6^6
into a number of simpler equations, according to non-divisibility by 7, 3 and 2.
Type A, t^6 = (6*x)^6 + (14*y)^6 + (21*w)^6 + (42*z3)^6 + (42*z4)^6 + (42*z5)^6 + (42*z6)^6
t mod 7 > 0, t mod 3 > 0, t mod 2 > 0
x mod 7 > 0
y mod 3 > 0
w mod 2 > 0
Type B, t^6 = (3*x)^6 + (14*y)^6 +(42*z2)^6 + (42*z3)^6 + (42*z4)^6 + (42*z5)^6 + (42*z6)^6
t mod 7 > 0, t mod 3 > 0, t mod 2 > 0
x mod 7 > 0, x mod 2 > 0
y mod 3 > 0
Type C, t^6 = (2*x)^6 + (21*y)^6 +(42*z2)^6 + (42*z3)^6 + (42*z4)^6 + (42*z5)^6 + (42*z6)^6
t mod 7 > 0, t mod 3 > 0, t mod 2 > 0
x mod 7 > 0, x mod 3 > 0
y mod 2 > 0
Type D, t^6 = (6*x)^6 + (7y)^6 + (42*z2)^6 + (42*z3)^6 + (42*z4)^6 + (42*z5)^6 + (42*z6)^6
t mod 7 > 0, t mod 3 > 0, t mod 2 > 0
x mod 7 > 0
y mod 3 > 0, y mod 2 > 0
Type E, t^6 = x^6 + (42*z1)^6 +(42*z2)^6 + (42*z3)^6 + (42*z4)^6 + (42*z5)^6 + (42*z6)^6
t mod 7 > 0, t mod 3 > 0, t mod 2 > 0
x mod 7 > 0, x mod 3 > 0, x mod 2 > 0
Type G, (7*t)^6 = (2*x)^6 + (3*y)^6 +(6*z2)^6 + (6*z3)^6 + (6*z4)^6 + (6*z5)^6 + (6*z6)^6
t mod 3 > 0, t mod 2 > 0
x mod 7 > 0, x mod 3 > 0
y mod 2 > 0
each z mod 7 > 0
Type H. (7*t)^6 = x^6 + (6*z1)^6 +(6*z2)^6 + (6*z3)^6 + (6*z4)^6 + (6*z5)^6 + (6*z6)^6
t mod 3 > 0, t mod 2 > 0
x mod 7 > 0, x mod 3 > 0, x mod 2 > 0
each z mod 7 > 0
To return from the simpler to the original equation, multiply each solution as follows.
A, 1, 6, 14, 21, 42, 42, 42, 42
B, 1, 3, 14, 42, 42, 42, 42, 42
C, 1, 2, 21, 42, 42, 42, 42, 42
D, 1, 6, 7, 42, 42, 42, 42, 42
E, 1, 1, 42, 42, 42, 42, 42, 42
G, 7, 2, 3, 6, 6, 6, 6, 6
H, 7, 1, 6, 6, 6, 6, 6, 6
Example
Using the smallest solution as an example,
1141^6 = 1077^6 + 894^6 + 702^6 + 474^6 + 402^6 + 234^6 + 74^6
Traditionally this result is presented in descending sequence, as
1141, 1077, 894, 702, 474, 402, 234, 74
However, this gives little insight into either the method of calculation or any underlying patterns.
This solution is type G,
163, 37, 359, 149, 117, 79, 67, 39
Multiplying by 7, 2, 3, 6, 6, 6, 6, 6 respectively gives
1141, 74, 1077, 894, 702, 474, 402, 234
Having analysed the seventy known solutions, I noticed that there were only seven published results for type G, and none for type H.
Although there is no possibility of serendipitously obtaining a (6,1,6) solution from these types, I see no good reason to ignore them.
All results are copywrite; in any reproduction, please acknowledge the discoverer of each solution.
Sums of seven sixth powers in tradition notation
Type H numerical solutions
t, x, z1, z2, z3, z4, z5, z6, who
7, 1, 6, 6, 6, 6, 6, 6, H
725, 3187, 705, 695, 656, 550, 309, 148, Peter J Ansell; Jan 2009
869, 2179, 968, 687, 680, 605, 446, 72, Peter J Ansell; Jan 2009
911, 1399, 1032, 726, 654, 451, 220, 174, Peter J Ansell; Jan 2009
1115, 29, 1234, 921, 911, 641, 631, 173, Peter J Ansell; Jan 2009
1123, 7691, 800, 764, 719, 512, 408, 85, Peter J Ansell; Jan 2009
1189, 547, 1233, 1172, 958, 675, 667, 632, Peter J Ansell; Jan 2009
1189, 547, 1282, 1069, 1028, 477, 314, 201, Peter J Ansell; Jan 2009
1207, 3103, 1394, 794, 752, 451, 93, 89, Peter J Ansell; Jan 2009
1261, 3323, 1280, 1250, 1107, 611, 381, 337, Peter J Ansell; Jan 2009
1267, 2309, 1228, 1210, 1181, 992, 761, 135, Peter J Ansell; Jan 2009
1267, 2309, 1433, 1089, 632, 578, 149, 44, Peter J Ansell; Jan 2009
1433, 7601, 1433, 1314, 1147, 974, 916, 747, Peter J Ansell; Jan 2009
1547, 3053, 1675, 1517, 769, 611, 471, 410, Peter J Ansell; Mar 2009
1549, 3067, 1698, 1457, 922, 869, 786, 121, Peter J Ansell; Mar 2009
1631, 4135, 1653, 1509, 1408, 1382, 850, 514, Peter J Ansell; Mar 2009
1651, 379, 1755, 1487, 1398, 1222, 789, 390, Peter J Ansell; Jan 2009
1759, 3239, 2045, 1068, 590, 285, 162, 30, Peter J Ansell; Jan 2009
1813, 12205, 1411, 1396, 1209, 806, 454, 414, Peter J Ansell; Jan 2009
2119, 719, 2216, 1860, 1755, 1720, 1536, 771, Peter J Ansell; Jan 2009
3737, 12721, 3882, 3616, 3165, 2076, 1797, 765, Peter J Ansell; Jan 2009
5269, 10259, 5742, 4770, 3898, 3750, 1692, 186, Peter J Ansell; Jan 2009 (hcf z = 2)
Type G numerical solutions
t, x, y, z2, z3, z4, z5, z6, who
7, 2, 3, 6, 6, 6, 6, 6, G
163, 37, 359, 149, 117, 79, 67, 39, Lander; Parkin and Selfridge 1966
235, 215, 433, 260, 144, 117, 103, 25, Jean-Charles Meyrignac 1999
251, 701, 65, 249, 208, 207, 200, 134, Jean-Charles Meyrignac 1999
287, 332, 115, 324, 251, 87, 33, 15, Jean-Charles Meyrignac 1999
289, 568, 115, 317, 258, 220, 40, 5, Jean-Charles Meyrignac 1999
353, 1114, 339, 326, 288, 278, 92, 85, Jean-Charles Meyrignac 1999
439, 1415, 587, 428, 241, 181, 166, 96, Jean-Charles Meyrignac 1999
539, 1279, 673, 580, 463, 426, 263, 230, Peter J Ansell; Jan 2009
569, 169, 1307, 422, 356, 116, 113, 13, Peter J Ansell; Jan 2009
635, 643, 927, 696, 561, 465, 264, 155, Peter J Ansell; Jan 2009
649, 1738, 459, 705, 545, 327, 235, 215, Peter J Ansell; Jan 2009
683, 1783, 1135, 737, 489, 397, 295, 116, Peter J Ansell; Jan 2009
691, 353, 727, 705, 701, 489, 488, 391, Peter J Ansell; Jan 2009
695, 124, 885, 806, 340, 286, 270, 173, Peter J Ansell; Jan 2009
697, 1103, 421, 801, 521, 362, 241, 3, Peter J Ansell; Jan 2009
709, 2014, 1121, 743, 542, 401, 74, 54, Peter J Ansell; Jan 2009
727, 1951, 1451, 685, 541, 529, 284, 15, Peter J Ansell; Jan 2009
737, 2458, 915, 663, 429, 312, 260, 111, Peter J Ansell; Jan 2009
739, 1423, 1063, 842, 510, 366, 276, 11, Peter J Ansell; Jan 2009
761, 860, 1147, 862, 564, 473, 100, 43, Peter J Ansell; Jan 2009
761, 1327, 165, 814, 652, 643, 550, 503, Peter J Ansell; Jan 2009
773, 2276, 929, 825, 502, 424, 223, 142, Peter J Ansell; Jan 2009
791, 998, 1611, 818, 538, 516, 275, 125, Peter J Ansell; Jan 2009
791, 1432, 725, 919, 306, 261, 66, 33, Peter J Ansell; Jan 2009
793, 1234, 1755, 639, 604, 582, 461, 290, Peter J Ansell; Jan 2009
823, 1129, 1707, 835, 591, 478, 296, 165, Peter J Ansell; Jan 2009
827, 1315, 1185, 918, 703, 541, 479, 16, Peter J Ansell; Jan 2009
841, 1607, 981, 943, 723, 452, 211, 116, Peter J Ansell; Jan 2009
859, 1003, 1023, 922, 755, 711, 634, 219, Peter J Ansell; Jan 2009
863, 746, 243, 948, 716, 674, 653, 461, Peter J Ansell; Jan 2009
863, 1441, 1613, 953, 415, 82, 72, 33, Peter J Ansell; Jan 2009
871, 983, 507, 1003, 617, 517, 394, 387, Peter J Ansell; Jan 2009
899, 352, 1447, 1027, 492, 241, 178, 143, Peter J Ansell; Jan 2009
917, 1772, 2097, 643, 619, 313, 250, 215, Peter J Ansell; Jan 2009
947, 2707, 489, 1028, 674, 411, 177, 6, Peter J Ansell; Jan 2009
959, 1868, 781, 1039, 888, 730, 110, 12, Peter J Ansell; Jan 2009
979, 1847, 41, 1101, 787, 717, 471, 177, Peter J Ansell; Jan 2009
991, 541, 461, 1054, 941, 807, 600, 225, Peter J Ansell; Jan 2009
1027, 2015, 1913, 1063, 929, 450, 436, 263, Peter J Ansell; Jan 2009
1037, 3022, 2057, 933, 789, 432, 122, 36, Peter J Ansell; Jan 2009
1043, 116, 681, 1207, 725, 435, 298, 85, Peter J Ansell; Jan 2009
1049, 148, 2203, 1039, 801, 589, 428, 395, Peter J Ansell; Jan 2009
1055, 1046, 1613, 1095, 1020, 814, 620, 363, Peter J Ansell; Jan 2009
1075, 3155, 1641, 1038, 997, 285, 228, 48, Peter J Ansell; Jan 2009
1081, 1198, 2395, 976, 674, 639, 594, 302, Peter J Ansell; Jan 2009
1081, 2413, 1989, 1158, 762, 736, 493, 416, Peter J Ansell; Jan 2009
1091, 1996, 409, 1228, 859, 809, 613, 326, Peter J Ansell; Jan 2009
1109, 1573, 1455, 1156, 1004, 984, 767, 11, Peter J Ansell; Jan 2009
1115, 136, 545, 1173, 1052, 920, 808, 108, Peter J Ansell; Jan 2009
1121, 3002, 1331, 1200, 881, 827, 720, 258, Peter J Ansell; Jan 2009
1129, 428, 309, 1303, 804, 622, 167, 116, Peter J Ansell; Jan 2009
1163, 2855, 2545, 1028, 638, 384, 249, 69, Peter J Ansell; Jan 2009
1189, 3250, 2271, 1224, 330, 299, 212, 108, Peter J Ansell; Jan 2009
1199, 4075, 541, 1014, 650, 593, 365, 360, Peter J Ansell; Jan 2009
1213, 2180, 2151, 1241, 992, 991, 960, 349, Peter J Ansell; Jan 2009
1249, 3769, 621, 1236, 1119, 690, 388, 215, Peter J Ansell; Jan 2009
1255, 832, 2085, 1423, 788, 544, 199, 52, Peter J Ansell; Jan 2009
1267, 2855, 585, 1338, 1228, 897, 213, 55, Peter J Ansell; Jan 2009
1295, 3608, 317, 1305, 1243, 782, 471, 82, Peter J Ansell; Jan 2009
1297, 773, 2845, 1117, 1097, 545, 436, 9, Peter J Ansell; Jan 2009
1303, 2122, 2581, 1389, 769, 755, 505, 433, Peter J Ansell; Mar 2009
1313, 143, 2253, 1398, 1226, 463, 389, 363, Peter J Ansell; Mar 2009
1319, 607, 2267, 1493, 659, 226, 206, 115, Peter J Ansell; Jan 2009
1337, 545, 1797, 1544, 748, 729, 395, 187, Peter J Ansell; Jan 2009
1375, 4057, 2605, 1298, 953, 856, 699, 242, Peter J Ansell; Jan 2009
1343, 2225, 2509, 1339, 1306, 426, 90, 87, Peter J Ansell; Mar 2009
1349, 2204, 1761, 1559, 732, 626, 352, 249, Peter J Ansell; Mar 2009
1385, 1081, 3179, 1048, 813, 617, 103, 51, Peter J Ansell; Jan 2009
1391, 3532, 131, 1553, 1067, 664, 481, 177, Peter J Ansell; Mar 2009
1397, 394, 1199, 1562, 1268, 465, 436, 207, Peter J Ansell; Jan 2009
1403, 4546, 449, 1347, 942, 870, 517, 41, Peter J Ansell; Jan 2009
1411, 1258, 1863, 1604, 1073, 859, 810, 174, Peter J Ansell; Mar 2009
1439, 4429, 2573, 1264, 1125, 1000, 857, 187, Peter J Ansell; Mar 2009
1457, 2069, 1699, 1535, 1481, 425, 309, 149, Peter J Ansell; Mar 2009
1477, 1999, 191, 1619, 1389, 873, 829, 716, Peter J Ansell; May 2009
1477, 3214, 2145, 1488, 1397, 1287, 820, 460, Peter J Ansell; May 2009
1487, 4597, 2403, 1497, 591, 452, 341, 18, Peter J Ansell; May 2009
1499, 4352, 3137, 1263, 823, 548, 324, 92, Peter J Ansell; May 2009
1501, 3616, 551, 1627, 1376, 869, 393, 29, Peter J Ansell; May 2009
1519, 1550, 2237, 1584, 1503, 1068, 755, 432, Peter J Ansell; May 2009
1523, 2293, 2391, 1510, 1416, 1317, 1215, 204, Peter J Ansell; May 2009
1537, 5015, 1811, 1427, 1052, 908, 892, 446, Peter J Ansell; May 2009
1553, 1490, 1437, 1734, 1380, 992, 752, 211, Peter J Ansell; June 2009
1573, 767, 991, 1660, 1447, 1289, 1189, 935, Peter J Ansell; June 2009
1573, 4822, 65, 1578, 1290, 981, 501, 142, Peter J Ansell; June 2009
1589, 4225, 1199, 1637, 1404, 1301, 905, 872, Peter J Ansell; June 2009
1595, 1087, 1697, 1796, 1401, 522, 179, 121, Peter J Ansell; June 2009
1597, 4495, 3207, 1380, 1370, 151, 100, 82, Peter J Ansell; June 2009
1601, 4996, 723, 1639, 1019, 838, 808, 362, Peter J Ansell; June 2009
1607, 2759, 1861, 1636, 1489, 1354, 1340, 730, Peter J Ansell; June 2009
1609, 5182, 373, 1356, 1348, 1258, 1002, 97, Peter J Ansell; June 2009
1613, 1037, 2633, 1800, 1312, 642, 515, 334, Peter J Ansell; June 2009
1613, 1280, 1367, 1692, 1635, 1024, 902, 594, Peter J Ansell; June 2009
1627, 956, 1343, 1858, 1285, 985, 704, 386, Peter J Ansell; June 2009
1649, 1276, 3741, 1283, 1224, 587, 116, 4, Peter J Ansell; June 2009
1645, 1262, 1559, 1727, 1650, 1202, 464, 229, Peter J Ansell; 01 July 2009
1655, 2026, 3541, 1641, 1048, 607, 498, 358, Peter J Ansell; June 2009
1657, 3005, 37, 1825, 1551, 776, 678, 188, Peter J Ansell; 02 July 2009
1669, 4505, 2623, 1790, 1297, 813, 398, 204, Peter J Ansell; June 2009
1673, 1117, 3573, 1578, 1371, 934, 562, 367, Peter J Ansell; June 2009
1675, 2278, 1681, 1871, 1373, 1324, 737, 45, Peter J Ansell; 03 July 2009
1679, 3811, 1725, 1924, 1020, 533, 363, 246, Peter J Ansell; 02 July 2009
1687, 1895, 1675, 1929, 1294, 1069, 438, 333, Peter J Ansell; 04 July 2009
Type E numerical solution
t, x, z1, z2, z3, z4, z5, z6, who
1, 1, 42, 42, 42, 42, 42, 42, E
244537, 231257, 4315, 3859, 3267, 2132, 1605, 1331, Rolan Christofferson 2007
Type D numerical solutions
t, x, y, z2, z3, z4, z5, z6, who
1, 6, 7, 42, 42, 42, 42, 42, D
34781, 5559, 2747, 580, 506, 440, 260, 195, Jean-Charles Meyrignac; Wannes de Smedt
38191, 1517, 4889, 803, 393, 296, 173, 148, Jean-Charles Meyrignac; Wannes de Smedt
108805, 12570, 11603, 2239, 2036, 1568, 1278, 409, Wannes de Smedt
127195, 1591, 8659, 2701, 2413, 2369, 348, 175, Greg Childers 2000
141709, 4010, 17525, 2719, 2675, 1902, 1790, 1242, Greg Childers 2000
145681, 4672, 9703, 3196, 2940, 1353, 1342, 605, Greg Childers 2000
177125, 16955, 1843, 3793, 3397, 3034, 2249, 426, Laurent Lucas 2000
208289, 5910, 10871, 4658, 4082, 780, 776, 137, Rolan Christofferson 2006
254941, 6927, 13499, 5866, 4557, 2400, 336, 206, Rolan Christofferson 2007
267803, 39992, 26797, 5386, 1975, 1690, 693, 91, Rolan Christofferson 2007
Type C numerical solutions
t, x, y, z2, z3, z4, z5, z6, who
1, 2, 21, 42, 42, 42, 42, 42, C
39779, 5917, 855, 893, 756, 524, 298, 70, Jean-Charles Meyrignac; Wannes de Smedt
66029, 12791, 2471, 1399, 1223, 926, 498, 292, Wannes de Smedt
194395, 40219, 1231, 4096, 3918, 3351, 1725, 448, Mark Dodrill 2000 (6;6;6)
194395, 40219, 3377, 4209, 3518, 3180, 2963, 2898, Mark Dodrill 2000
201133, 80762, 441, 4511, 2454, 2379, 761, 477, Jean-Charles Meyrignac 2000
212405, 66692, 449, 4567, 4194, 3140, 2346, 1994, Rolan Christofferson 2006
236323, 38452, 5257, 5416, 4209, 2657, 2281, 2058, Rolan Christofferson 2006
283501, 96188, 13113, 3901, 3262, 3252, 1141, 222, Rolan Christofferson 2007
296917, 145664, 3583, 4764, 3431, 2140, 846, 601, Rolan Christofferson
297329, 89840, 12499, 6043, 4718, 2734, 1605, 1164, Rolan Christofferson
Type B numerical solutions
t, x, y, z2, z3, z4, z5, z6, who
1, 3, 14, 42, 42, 42, 42, 42, B
125945, 27149, 5879, 2771, 2078, 2040, 1492, 991, Greg Childers 2000
130783, 28555, 7172, 2865, 1977, 1715, 1564, 695, Greg Childers 2000
172129, 49067, 4906, 3704, 2558, 1099, 992, 906, Greg Childers 2000
198689, 12203, 12682, 4057, 3115, 2243, 940, 926, Mark Dodrill 2000
204737, 10187, 14572, 2340, 2180, 1422, 1134, 1129, Rolan Christofferson 2006
239147, 79257, 5222, 3191, 1828, 1674, 1601, 113, Rolan Christofferson 2006
248743, 80403, 232, 3818, 3779, 3053, 2721, 1894, Rolan Christofferson 2007
305825, 23509, 7541, 2305, 6752, 5811, 4934, 3132, Rolan Christofferson
Type A numerical solutions
t, x, y, w, z3, z4, z5, z6, who
1, 6, 14, 21, 42, 42, 42, 42, A
54347, 5458, 2384, 927, 1163, 1096, 75, 50, Wannes de Smedt
59819, 1567, 541, 415, 1232, 1151, 1055, 1023, Wannes de Smedt
63631, 9003, 2792, 1203, 1289, 1145, 554, 32, Wannes de Smedt
67681, 8328, 200, 381, 1550, 847, 789, 763, Wannes de Smedt
70513, 9663, 4325, 877, 1363, 620, 516, 317, Wannes de Smedt
78919, 6455, 4135, 1941, 1809, 919, 622, 328, Wannes de Smedt
89797, 4642, 4175, 559, 1990, 1612, 1429, 239, Wannes de Smedt
97081, 3428, 6188, 1781, 1941, 1665, 756, 519, Wannes de Smedt (hcf z = 3)
123943, 1049, 7586, 5323, 1741, 1581, 701, 302, Greg Childers 2000
144175, 2742, 8198, 5133, 3126, 985, 275, 144, Greg Childers 2000
152041, 5732, 10565, 5115, 1945, 1397, 995, 738, Greg Childers 2000
171971, 4843, 6896, 7927, 2939, 1851, 1215, 196, Laurent Lucas 2000
173905, 27795, 5161, 1261, 3103, 2220, 2083, 83, Laurent Lucas 2000
185659, 8536, 9599, 6193, 4135, 2800, 1682, 84, Greg Childers 2000
204497, 29545, 3080, 1491, 4007, 3902, 712, 215, Mark Dodrill 2000
204499, 21327, 9016, 3385, 4245, 4063, 3253, 2474, Mark Dodrill 2000
210215, 661, 6143, 9813, 3453, 835, 826, 83, Rolan Christofferson 2006
221959, 8912, 3548, 4725, 5114, 3889, 2512, 1276, Rolan Christofferson 2006
227653, 33911, 5864, 8271, 4412, 630, 334, 103, Rolan Christofferson 2006
227959, 34159, 11336, 4059, 4549, 2382, 1649, 1599, Rolan Christofferson 2006
234571, 1592, 4553, 511, 5260, 4524, 2591, 2561, Rolan Christofferson 2006
244147, 21083, 15661, 8825, 4555, 3135, 2624, 1485, Rolan Christofferson 2007
244463, 974, 7550, 6579, 5761, 3028, 1807, 312, Rolan Christofferson 2007
249031, 12707, 17638, 725, 3578, 1543, 1230, 968, Rolan Christofferson 2007
249251, 22625, 9395, 2903, 5788, 3837, 3048, 89, Rolan Christofferson 2007
250079, 24600, 17459, 3811, 3940, 1723, 1404, 1354, Rolan Christofferson 2007
262819, 12606, 18443, 1449, 4203, 2819, 1422, 301, Rolan Christofferson 2007
268843, 25199, 1412, 10305, 5979, 3583, 2110, 1439, Rolan Christofferson 2007
270715, 25511, 6857, 11887, 5296, 3814, 924, 9, Rolan Christofferson 2007
271423, 25629, 11977, 9667, 5912, 4492, 3736, 124, Rolan Christofferson 2007 (hcf z = 4)
279073, 30764, 13444, 6627, 5810, 5550, 3469, 1071, Rolan Christofferson 2007
289549, 28650, 3863, 10905, 5583, 5252, 5105, 4415, Rolan Christofferson
294955, 28201, 8734, 1791, 6582, 5667, 2892, 1966, Rolan Christofferson
307553, 34498, 17542, 12189, 5973, 3589, 3207, 1121, Rolan Christofferson
Clearly, there are many other ways to catalogue these solutions.
This data is presented "as is" without warranty; please let me know of any errors or omissions.
My email is xxxx at computer-man.demon.co.uk where math replaces xxxx.
Sums of seven sixth powers in ascending sequence
Diophantine equations with one solution
Equal Sums of Fourth Powers: Biquadratic solutions to (4,2,2) and (4,1,4) from Jaroslaw Wroblewski
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