Diophantine Equation--8th Powers
The 2-1 equation
 | (1) |
, and so has no solution. No 2-2 solutions are known.No 3-1, 3-2, or 3-3 solutions are known.
No 4-1, 4-2, 4-3, or 4-4 solutions are known.
No 5-1, 5-2, 5-3, or 5-4 solutions are known, but Letac (1942) found a solution to the 5-5 equation. The smallest5-5 solution is
 | (2) |
No 6-1, 6-2, 6-3, or 6-4 solutions are known. Moessner and Gloden (1944) found solutions to the 6-6 equation. Thesmallest 6-6 solution is
 | (3) |
No 7-1, 7-2, or 7-3 solutions are known. The smallest 7-4 solution is
 | (4) |
 | (5) |
No 8-1 or 8-2 solutions are known. The smallest 8-3 solution is
 | (6) |
 | |
 | (7) |
No solutions to the 9-1 equation are known. The smallest 9-2 solution is
 | (8) |
No solutions to the 10-1 equation are known.
The smallest 11-1 solution is
 | (9) |
The smallest 12-1 solution is
 | (10) |
The general identity
 | (11) |
References
Gloden, A. ``Parametric Solutions of Two Multi-Degreed Equalities.'' Amer. Math. Monthly 55, 86-88, 1948. Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. ``A Survey of Equal Sums of Like Powers.'' Math. Comput. 21, 446-459, 1967. Letac, A. Gazetta Mathematica 48, 68-69, 1942. Moessner, A. ``On Equal Sums of Like Powers.'' Math. Student 15, 83-88, 1947. Moessner, A. and Gloden, A. ``Einige Zahlentheoretische Untersuchungen und Resultante.'' Bull. Sci. École Polytech. de Timisoara 11, 196-219, 1944. Sastry, S. ``On Sums of Powers.'' J. London Math. Soc. 9, 242-246, 1934.
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