## Diophantine Equation--8th Powers

The 2-1 equation

 (1)

is a special case of Fermat's Last Theorem with , 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 smallest 5-5 solution is

 (2)

(Lander et al. 1967).

No 6-1, 6-2, 6-3, or 6-4 solutions are known. Moessner and Gloden (1944) found solutions to the 6-6 equation. The smallest 6-6 solution is

 (3)

(Lander et al. 1967).

No 7-1, 7-2, or 7-3 solutions are known. The smallest 7-4 solution is

 (4)

(Lander et al. 1967). Moessner and Gloden (1944) found solutions to the 7-6 equation. Parametric solutions to the 7-7 equation were given by Moessner (1947) and Gloden (1948). The smallest 7-7 solution is

 (5)

(Lander et al. 1967).

No 8-1 or 8-2 solutions are known. The smallest 8-3 solution is

 (6)

(Lander et al. 1967). Sastry (1934) used the smallest 17-1 solution to give a parametric 8-8 solution. The smallest 8-8 solution is
 (7)
(Lander et al. 1967).

No solutions to the 9-1 equation are known. The smallest 9-2 solution is

 (8)

(Lander et al. 1967). Letac (1942) found solutions to the 9-9 equation.

No solutions to the 10-1 equation are known.

The smallest 11-1 solution is

 (9)

(Lander et al. 1967).

The smallest 12-1 solution is

 (10)

(Lander et al. 1967).

The general identity

 (11)

gives a solution to the 17-1 equation (Lander et al. 1967).

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.