## Diophantine Equation--9th Powers

The 2-1 equation

 (1)

is a special case of Fermat's Last Theorem with , and so has no solution. There is no known 2-2 solution.

There are no known 3-1, 3-2, or 3-3 solutions.

There are no known 4-1, 4-2, 4-3, or 4-4 solutions.

There are no known 5-1, 5-2, 5-3, 5-4, or 5-5 solutions.

There are no known 6-1, 6-2, 6-3, 6-4, or 6-5 solutions. The smallest 6-6 solution is

 (2)

(Lander et al. 1967).

There are no known 7-1, 7-2, 7-3, 7-4, or 7-5 solutions.

There are no known 8-1, 8-2, 8-3, 8-4, or 8-5 solutions.

There are no known 9-1, 9-2, 9-3, 9-4, or 9-5 solutions.

There are no known 10-1, 10-2, or 10-3 solutions. The smallest 10-4 solution is

 (3)

(Lander et al. 1967). No 10-5 solution is known. Moessner (1947) gives a parametric solution to the 10-10 equation.

There are no known 11-1 or 11-2 solutions. The smallest 11-3 solution is

 (4)

(Lander et al. 1967). The smallest 11-5 solution is
 (5)
(Lander et al. 1967). Palamá (1953) gave a solution to the 11-11 equation.

There is no known 12-1 solution. The smallest 12-2 solution is

 (6)

(Lander et al. 1967).

There are no known 13-1 or 14-1 solutions. The smallest 15-1 solution is
 (7)
(Lander et al. 1967).

References

Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. A Survey of Equal Sums of Like Powers.'' Math. Comput. 21, 446-459, 1967.

Moessner, A. On Equal Sums of Like Powers.'' Math. Student 15, 83-88, 1947.

Palamá, G. Diophantine Systems of the Type (, 2, ..., , , , ..., ).'' Scripta Math. 19, 132-134, 1953.