k=2,3,4,... x^k=a+b, y^k=a^2+b^2

王守恩

x^2 = a+b,  y^2 = a^2+b^2——非平凡解可以有无穷。约定a≤b。
非平凡解{x, y, a, b}={{1, 985, -696, 697}, {7, 1069, -731, 780}, {17, 901, -476, 765}, {23, 1495, -759, 1288}, {31, 1745, -656, 1617}, {41, 1741, -59, 1740}, {47, 3055, -752, 2961}, ...}
x={1, 7, 17, 23, 31, 41, 47, 49, 71, 73, 79, 89, 97, 103, 113, 119, 127, 137, 151, 161, 167, 191, 193, 199, 217, 223, 233, 239, 241, 257, 263, 271, 281, 287, 289, 311, 313, 329, 337, ...

x^3 = a+b,  y^3 = a^2+b^2——非平凡解可以有无穷。约定a≤b。
非平凡解{x, y, a, b}={{1, 1, 0, 1}, {2, 10, -18, 26}, {3, 53, -259, 286}, {44, 1570, 31598, 53586}, {61, 75661, -14602185, 14829166}, {3270, 16090058, -24673078334, 59638861334}, ...}
x={1, 2, 3, 44, 61, 3270, 44239, 483736, 70318863, 3590624822, 374582200541, 137850021357180, 1052807601737059, 16316195654413260658, 5995647162749122514817, ...

x^4 = a+b,  y^4 = a^2+b^2——非平凡解只有2个。{x, y, a, b}={1, 13, -119, 120}, {1, 1, 0, 1}。约定a≤b。
平凡解{x, y, a, b}={1, 13, -119, 120}。{2, 52, -1904, 1920}, {3, 117, -9639, 9720}, {4, 208, -30464, 30720}, {5, 325, -74375, 75000}, {6, 468, -154224, 155520}, ...}
平凡解{x, y, a, b}={{1, 1, 0, 1}, {2, 4, 0, 16}, {3, 9, 0, 81}, {4, 16, 0, 256}, {5, 25, 0, 625}, {6, 36, 0, 1296}, {7, 49, 0, 2401}, {8, 64, 0, 4096}, {9, 81, 0, 6561}, ...}

x^5 = a+b,  y^5 = a^2+b^2——非平凡解只有1个。{x, y, a, b}={1,1,0,1}。约定a≤b。
平凡解{x, y, a, b}={{1, 1, 0, 1}, {2, 4, 0, 32}, {3, 9, 0, 243}, {4, 16, 0, 1024}, {5, 25, 0, 3125}, {6, 36, 0, 7776}, {7, 49, 0, 16807}, {8, 64, 0, 32768}, {9, 81, 0, 59049}, ...}

x^6 = a+b,  y^6 = a^2+b^2——非平凡解只有1个。{x, y, a, b}={1,1,0,1}。约定a≤b。
平凡解{x, y, a, b}={{1, 1, 0, 1}, {2, 4, 0, 64}, {3, 9, 0, 729}, {4, 16, 0, 4096}, {5, 25, 0, 15625}, {6, 36, 0, 46656}, {7, 49, 0, 117649}, {8, 64, 0, 262144}, {9, 81, 0, 531441}, ...}

x^7 = a+b,  y^7 = a^2+b^2——非平凡解只有1个。{x, y, a, b}={1,1,0,1}。约定a≤b。
平凡解{x, y, a, b}={{1, 1, 0, 1}, {2, 4, 0, 128}, {3, 9, 0, 2187}, {4, 16, 0, 16384}, {5, 25, 0, 78125}, {6, 36, 0, 279936}, {7, 49, 0, 823543}, {8, 64, 0, 2097152}, {9, 81, 0, 4782969}, ...}

x^8 = a+b,  y^8 = a^2+b^2——非平凡解只有1个。{x, y, a, b}={1,1,0,1}。约定a≤b。
平凡解{x, y, a, b}={{1, 1, 0, 1}, {2, 4, 0, 256}, {3, 9, 0, 6561}, {4, 16, 0, 65536}, {5, 25, 0, 390625}, {6, 36, 0, 1679616}, {7, 49, 0, 5764801}, {8, 64, 0, 16777216}, {9, 81, 0, 43046721}, ...}

x^9 = a+b,  y^9 = a^2+b^2——非平凡解只有1个。{x, y, a, b}={1,1,0,1}。约定a≤b。
平凡解{x, y, a, b}={{1, 1, 0, 1}, {2, 4, 0, 512}, {3, 9, 0, 19683}, {4, 16, 0, 262144}, {5, 25, 0, 1953125}, {6, 36, 0, 10077696}, {7, 49, 0, 40353607}, {8, 64, 0, 134217728}, {9, 81, 0, 387420489}, ...}