Diophantine Equation--9th Powers

The 2-1 equation

$$A^9+B^9=C^9$$ (1)

is a special case of Fermat's Last Theorem with $n=9$, 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

$$1^9+13^9+13^9+14^9+18^9+23^9 = 5^9+9^9+10^9+15^9+21^9+22^9$$ (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

$$2^9+6^9+6^9+9^9+10^9+11^9+14^9+18^9+19^9+19^9 = 5^9+12^9+16^9+21^9$$ (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

$$2^9+3^9+6^9+7^9+9^9+9^9+19^9+19^9+21^9+25^9+29^9=13^9+16^9+30^9$$ (4)

(Lander et al. 1967). The smallest 11-5 solution is

$3^9+5^9+5^9+9^9+9^9+12^9+15^9+15^9+16^9+21^9+21^9$
$=7^9+8^9+14^9+20^9+22^9\quad$	(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

$$4\cdot 2^9+2\cdot 3^9+4^9+7^9+16^9+17^9+2\cdot 19^9 = 15^9+21^9$$ (6)

(Lander et al. 1967).

There are no known 13-1 or 14-1 solutions. The smallest 15-1 solution is

$2^9+2^9+4^9+6^9+6^9+7^9+9^9+9^9+10^9+15^9$
$+18^9+21^9+21^9+23^9+23^9=26^9\quad$	(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 $\sum_{i=1}^p {a_i}^k=\sum_{i=1}^p {b_i}^k$ ($k=1$, 2, ..., $n$, $n+2$, $n+4$, ..., $n+2r$).'' Scripta Math. 19, 132-134, 1953.