// We list the solutions // to x^2+2^{alpha_2} 3^{alpha_3} 5^{alpha_5} 7^{alpha_7} 11^{alpha_11} = y^n // with x>0, gcd(x,y)=1 and n \ge 3. // The solutions are given in the format [n, x, y, alpha_2, alpha_3, alpha_5, alpha_7, alpha_11]. sols:= [ [ 3, 1, 2, 0, 0, 0, 1, 0 ], [ 3, 1, 4, 0, 2, 0, 1, 0 ], [ 3, 2, 5, 0, 0, 0, 0, 2 ], [ 3, 3, 4, 0, 0, 1, 0, 1 ], [ 3, 4, 3, 0, 0, 0, 0, 1 ], [ 3, 5, 3, 1, 0, 0, 0, 0 ], [ 3, 5, 9, 6, 0, 0, 0, 1 ], [ 3, 7, 4, 0, 1, 1, 0, 0 ], [ 3, 9, 5, 2, 0, 0, 0, 1 ], [ 3, 9, 11, 1, 0, 4, 0, 0 ], [ 3, 10, 7, 0, 5, 0, 0, 0 ], [ 3, 11, 5, 2, 0, 0, 0, 0 ], [ 3, 13, 8, 0, 0, 0, 3, 0 ], [ 3, 13, 9, 4, 0, 1, 1, 0 ], [ 3, 17, 7, 1, 3, 0, 0, 0 ], [ 3, 17, 9, 3, 0, 1, 0, 1 ], [ 3, 17, 97, 10, 4, 0, 0, 1 ], [ 3, 22, 9, 0, 0, 1, 2, 0 ], [ 3, 23, 9, 3, 0, 2, 0, 0 ], [ 3, 28, 19, 0, 5, 2, 0, 0 ], [ 3, 29, 11, 1, 0, 1, 2, 0 ], [ 3, 29, 25, 6, 1, 0, 1, 1 ], [ 3, 34, 11, 0, 0, 2, 1, 0 ], [ 3, 35, 13, 2, 5, 0, 0, 0 ], [ 3, 36, 11, 0, 0, 1, 1, 0 ], [ 3, 37, 14, 0, 0, 3, 0, 1 ], [ 3, 37, 25, 4, 4, 0, 0, 1 ], [ 3, 41, 15, 1, 0, 0, 1, 2 ], [ 3, 46, 13, 0, 4, 0, 0, 0 ], [ 3, 53, 19, 1, 4, 2, 0, 0 ], [ 3, 55, 37, 2, 5, 0, 2, 0 ], [ 3, 57, 16, 0, 0, 0, 1, 2 ], [ 3, 58, 15, 0, 0, 0, 0, 1 ], [ 3, 61, 16, 0, 1, 3, 0, 0 ], [ 3, 71, 25, 3, 3, 0, 2, 0 ], [ 3, 71, 31, 1, 2, 3, 0, 1 ], [ 3, 82, 19, 0, 3, 1, 0, 0 ], [ 3, 83, 29, 2, 0, 4, 1, 0 ], [ 3, 89, 31, 1, 7, 1, 0, 0 ], [ 3, 91, 61, 2, 7, 2, 0, 0 ], [ 3, 107, 37, 2, 4, 0, 0, 2 ], [ 3, 116, 31, 0, 3, 1, 0, 2 ], [ 3, 117, 25, 4, 0, 0, 0, 2 ], [ 3, 118, 25, 0, 5, 0, 1, 0 ], [ 3, 119, 71, 1, 0, 6, 0, 1 ], [ 3, 136, 91, 0, 5, 2, 0, 2 ], [ 3, 143, 49, 4, 5, 2, 0, 0 ], [ 3, 153, 29, 2, 0, 1, 2, 0 ], [ 3, 154, 31, 0, 5, 2, 0, 0 ], [ 3, 170, 31, 0, 4, 0, 0, 1 ], [ 3, 171, 31, 1, 0, 2, 0, 1 ], [ 3, 173, 34, 0, 1, 5, 0, 0 ], [ 3, 181, 32, 0, 0, 0, 1, 0 ], [ 3, 188, 37, 0, 7, 0, 1, 0 ], [ 3, 190, 37, 0, 3, 0, 2, 1 ], [ 3, 197, 34, 0, 2, 1, 0, 1 ], [ 3, 197, 67, 1, 5, 0, 2, 1 ], [ 3, 217, 37, 2, 4, 0, 0, 1 ], [ 3, 223, 37, 2, 1, 0, 1, 1 ], [ 3, 225, 37, 2, 0, 0, 1, 0 ], [ 3, 233, 49, 7, 2, 1, 0, 1 ], [ 3, 239, 81, 4, 0, 1, 2, 2 ], [ 3, 251, 85, 2, 9, 0, 1, 0 ], [ 3, 253, 169, 4, 5, 2, 2, 0 ], [ 3, 260, 43, 0, 5, 0, 2, 0 ], [ 3, 261, 41, 5, 0, 2, 0, 0 ], [ 3, 262, 49, 0, 4, 1, 0, 2 ], [ 3, 281, 46, 0, 1, 3, 2, 0 ], [ 3, 307, 46, 0, 2, 0, 3, 0 ], [ 3, 323, 109, 2, 5, 2, 2, 0 ], [ 3, 332, 49, 0, 3, 2, 0, 1 ], [ 3, 377, 235, 1, 5, 0, 4, 1 ], [ 3, 383, 129, 7, 0, 6, 0, 0 ], [ 3, 419, 56, 0, 0, 1, 0, 1 ], [ 3, 433, 58, 0, 2, 0, 1, 2 ], [ 3, 433, 289, 8, 5, 1, 1, 1 ], [ 3, 440, 67, 0, 7, 0, 2, 0 ], [ 3, 442, 79, 0, 5, 2, 2, 0 ], [ 3, 449, 151, 1, 3, 2, 4, 0 ], [ 3, 476, 61, 0, 4, 1, 0, 0 ], [ 3, 496, 331, 0, 5, 2, 2, 2 ], [ 3, 503, 169, 3, 3, 2, 1, 2 ], [ 3, 505, 67, 1, 3, 0, 1, 2 ], [ 3, 524, 65, 0, 0, 0, 2, 0 ], [ 3, 538, 67, 0, 1, 0, 3, 1 ], [ 3, 547, 169, 4, 1, 2, 3, 1 ], [ 3, 568, 109, 0, 4, 1, 4, 0 ], [ 3, 575, 193, 6, 7, 0, 2, 0 ], [ 3, 595, 73, 4, 7, 0, 0, 0 ], [ 3, 596, 71, 0, 0, 1, 2, 1 ], [ 3, 603, 137, 12, 0, 0, 2, 1 ], [ 3, 629, 1831, 1, 13, 2, 1, 1 ], [ 3, 665, 97, 4, 5, 0, 0, 2 ], [ 3, 671, 141, 2, 0, 1, 6, 0 ], [ 3, 679, 81, 8, 0, 2, 0, 1 ], [ 3, 685, 79, 1, 5, 0, 2, 0 ], [ 3, 692, 79, 0, 4, 2, 1, 0 ], [ 3, 797, 86, 0, 0, 0, 1, 2 ], [ 3, 813, 89, 5, 0, 3, 0, 1 ], [ 3, 835, 89, 6, 0, 0, 0, 2 ], [ 3, 836, 91, 0, 7, 2, 0, 0 ], [ 3, 837, 89, 4, 0, 2, 0, 1 ], [ 3, 865, 91, 1, 5, 0, 0, 1 ], [ 3, 874, 151, 0, 7, 2, 2, 0 ], [ 3, 899, 301, 2, 7, 2, 0, 2 ], [ 3, 908, 97, 0, 6, 0, 0, 2 ], [ 3, 910, 103, 0, 7, 0, 0, 2 ], [ 3, 918, 449, 0, 0, 3, 2, 4 ], [ 3, 955, 97, 3, 4, 0, 0, 0 ], [ 3, 974, 163, 0, 1, 0, 1, 5 ], [ 3, 1009, 136, 0, 2, 3, 0, 3 ], [ 3, 1043, 115, 1, 9, 0, 0, 1 ], [ 3, 1091, 106, 0, 1, 1, 2, 0 ], [ 3, 1133, 109, 2, 4, 1, 1, 0 ], [ 3, 1133, 379, 1, 5, 6, 1, 0 ], [ 3, 1135, 109, 2, 5, 0, 1, 0 ], [ 3, 1169, 121, 3, 4, 4, 0, 0 ], [ 3, 1261, 121, 6, 4, 1, 1, 0 ], [ 3, 1293, 119, 1, 0, 1, 0, 3 ], [ 3, 1315, 121, 5, 3, 0, 2, 0 ], [ 3, 1387, 169, 10, 4, 1, 1, 0 ], [ 3, 1436, 141, 0, 0, 3, 2, 2 ], [ 3, 1450, 247, 0, 7, 0, 2, 2 ], [ 3, 1495, 133, 2, 5, 0, 0, 2 ], [ 3, 1583, 529, 4, 3, 4, 2, 1 ], [ 3, 1700, 163, 0, 5, 0, 2, 2 ], [ 3, 1729, 181, 2, 5, 2, 0, 2 ], [ 3, 1819, 421, 2, 3, 2, 4, 1 ], [ 3, 1861, 841, 12, 1, 4, 1, 1 ], [ 3, 2113, 184, 0, 1, 1, 6, 0 ], [ 3, 2171, 201, 9, 0, 1, 0, 3 ], [ 3, 2177, 169, 3, 7, 1, 0, 0 ], [ 3, 2242, 379, 0, 3, 3, 0, 4 ], [ 3, 2353, 177, 4, 0, 0, 2, 1 ], [ 3, 2429, 181, 2, 3, 2, 0, 1 ], [ 3, 2512, 289, 0, 3, 2, 4, 1 ], [ 3, 2629, 241, 3, 11, 1, 0, 0 ], [ 3, 2681, 193, 4, 4, 0, 0, 0 ], [ 3, 2701, 1801, 4, 5, 4, 4, 0 ], [ 3, 2771, 331, 1, 3, 4, 1, 2 ], [ 3, 2801, 201, 3, 0, 5, 0, 1 ], [ 3, 2807, 199, 1, 3, 2, 0, 0 ], [ 3, 2815, 221, 2, 0, 0, 2, 4 ], [ 3, 2951, 301, 2, 3, 6, 0, 1 ], [ 3, 3016, 211, 0, 5, 2, 2, 0 ], [ 3, 3059, 211, 1, 6, 2, 0, 0 ], [ 3, 3093, 478, 0, 0, 0, 7, 2 ], [ 3, 3403, 2269, 2, 9, 2, 2, 2 ], [ 3, 3419, 313, 4, 4, 0, 0, 4 ], [ 3, 3437, 229, 2, 4, 1, 0, 2 ], [ 3, 3443, 249, 12, 0, 3, 1, 0 ], [ 3, 3448, 229, 0, 7, 1, 0, 1 ], [ 3, 3689, 241, 6, 5, 2, 0, 0 ], [ 3, 3781, 415, 1, 5, 0, 6, 0 ], [ 3, 3835, 289, 3, 7, 0, 2, 1 ], [ 3, 3893, 309, 2, 0, 1, 2, 4 ], [ 3, 4024, 253, 0, 5, 0, 1, 0 ], [ 3, 4069, 265, 8, 6, 0, 0, 1 ], [ 3, 4085, 553, 6, 9, 0, 0, 2 ], [ 3, 4229, 284, 0, 0, 0, 3, 4 ], [ 3, 4292, 379, 0, 5, 2, 2, 2 ], [ 3, 4318, 1499, 0, 0, 10, 3, 0 ], [ 3, 4843, 1177, 4, 15, 0, 1, 0 ], [ 3, 4859, 1621, 2, 6, 1, 4, 2 ], [ 3, 4879, 481, 6, 7, 4, 0, 0 ], [ 3, 4888, 289, 0, 4, 2, 0, 2 ], [ 3, 4964, 291, 0, 0, 3, 1, 0 ], [ 3, 5066, 295, 0, 6, 0, 0, 1 ], [ 3, 5291, 361, 6, 5, 2, 2, 0 ], [ 3, 5356, 331, 0, 9, 1, 1, 1 ], [ 3, 5491, 313, 6, 6, 0, 0, 1 ], [ 3, 5497, 785, 8, 0, 0, 0, 6 ], [ 3, 5597, 1369, 13, 2, 5, 0, 1 ], [ 3, 5797, 337, 3, 5, 0, 4, 0 ], [ 3, 5805, 323, 1, 0, 0, 0, 2 ], [ 3, 5993, 379, 1, 7, 1, 1, 2 ], [ 3, 6021, 331, 1, 0, 3, 2, 0 ], [ 3, 6049, 793, 3, 7, 0, 4, 1 ], [ 3, 6179, 345, 18, 0, 0, 0, 1 ], [ 3, 6203, 844, 0, 1, 7, 4, 0 ], [ 3, 6616, 361, 0, 5, 2, 2, 1 ], [ 3, 6706, 421, 0, 4, 2, 0, 4 ], [ 3, 6749, 361, 4, 5, 1, 1, 1 ], [ 3, 7372, 379, 0, 5, 1, 1, 1 ], [ 3, 7429, 531, 1, 0, 8, 0, 2 ], [ 3, 7568, 499, 0, 7, 4, 2, 0 ], [ 3, 7811, 394, 0, 2, 0, 5, 0 ], [ 3, 7910, 397, 0, 5, 0, 0, 1 ], [ 3, 7911, 401, 6, 0, 1, 2, 2 ], [ 3, 8092, 949, 0, 15, 1, 0, 1 ], [ 3, 8099, 433, 3, 11, 0, 0, 1 ], [ 3, 8173, 409, 5, 4, 4, 0, 0 ], [ 3, 8441, 421, 2, 7, 1, 1, 1 ], [ 3, 8569, 421, 2, 5, 2, 2, 0 ], [ 3, 9143, 709, 2, 11, 1, 1, 1 ], [ 3, 9324, 443, 0, 0, 0, 0, 3 ], [ 3, 9703, 529, 6, 7, 1, 1, 1 ], [ 3, 9757, 457, 4, 7, 0, 1, 0 ], [ 3, 9890, 463, 0, 5, 0, 2, 2 ], [ 3, 9959, 465, 10, 0, 0, 0, 3 ], [ 3, 10261, 1171, 1, 11, 1, 1, 2 ], [ 3, 10387, 589, 2, 9, 2, 2, 0 ], [ 3, 10513, 609, 21, 0, 1, 0, 1 ], [ 3, 10549, 481, 3, 4, 1, 0, 0 ], [ 3, 11026, 7351, 0, 7, 4, 4, 2 ], [ 3, 11143, 499, 1, 5, 2, 1, 0 ], [ 3, 11339, 541, 2, 5, 4, 2, 0 ], [ 3, 11341, 505, 7, 3, 0, 2, 0 ], [ 3, 11395, 793, 8, 5, 0, 2, 2 ], [ 3, 11843, 529, 3, 4, 1, 4, 0 ], [ 3, 12502, 559, 0, 5, 4, 0, 2 ], [ 3, 13096, 631, 0, 6, 6, 1, 0 ], [ 3, 13271, 561, 8, 0, 1, 3, 0 ], [ 3, 13328, 619, 0, 9, 2, 0, 2 ], [ 3, 13490, 1807, 0, 9, 0, 4, 2 ], [ 3, 13537, 568, 0, 2, 0, 1, 0 ], [ 3, 14651, 601, 4, 5, 4, 0, 0 ], [ 3, 15023, 609, 7, 0, 3, 0, 1 ], [ 3, 15344, 661, 0, 6, 1, 0, 4 ], [ 3, 15821, 631, 1, 5, 2, 1, 1 ], [ 3, 16103, 639, 1, 0, 1, 0, 5 ], [ 3, 16523, 649, 5, 7, 1, 0, 0 ], [ 3, 16757, 655, 1, 7, 0, 2, 0 ], [ 3, 16927, 709, 2, 1, 4, 1, 3 ], [ 3, 17303, 669, 2, 0, 3, 2, 0 ], [ 3, 17342, 679, 0, 9, 4, 0, 0 ], [ 3, 17459, 673, 3, 4, 0, 1, 0 ], [ 3, 17641, 961, 4, 5, 2, 2, 2 ], [ 3, 17771, 681, 9, 0, 2, 0, 0 ], [ 3, 18073, 709, 2, 5, 4, 2, 0 ], [ 3, 18395, 697, 4, 3, 0, 2, 1 ], [ 3, 18649, 751, 1, 9, 2, 1, 1 ], [ 3, 18863, 709, 2, 5, 1, 0, 2 ], [ 3, 18880, 751, 0, 1, 0, 5, 3 ], [ 3, 19711, 991, 1, 5, 6, 1, 1 ], [ 3, 20069, 1021, 2, 7, 4, 0, 2 ], [ 3, 20213, 889, 4, 5, 4, 0, 2 ], [ 3, 20735, 757, 2, 9, 0, 2, 0 ], [ 3, 20791, 793, 3, 4, 0, 1, 4 ], [ 3, 21104, 781, 0, 11, 2, 1, 0 ], [ 3, 22331, 793, 4, 1, 0, 1, 1 ], 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