Diophantine Equation--8th Powers

The 2-1 equation

$\begin{displaymath} A^8+B^8=C^8 \end{displaymath}$

(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

$\begin{displaymath} 1^8+10^8+11^8+20^8+43^8=5^8+28^8+32^8+35^8+41^8 \end{displaymath}$

(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

$\begin{displaymath} 3^8+6^8+8^8+10^8+15^8+23^8=5^8+9^8+9^8+12^8+20^8+22^8 \end{displaymath}$

(3)

(Lander et al. 1967).

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

$\begin{displaymath} 7^8+9^8+16^8+22^8+22^8+28^8+34^8=6^8+11^8+20^8+35^8 \end{displaymath}$

(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

$\begin{displaymath} 1^8+3^8+5^8+6^8+6^8+8^8+13^8 = 4^8+7^8+9^8+9^8+10^8+11^8+12^8 \end{displaymath}$

(5)

(Lander et al. 1967).

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

$\begin{displaymath} 6^8+12^8+16^8+16^8+38^8+38^8+40^8+47^8=8^8+17^8+50^8 \end{displaymath}$

(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


$=4^8+5^8+5^8+6^8+6^8+11^8+11^8+11^8\quad$	(7)

(Lander et al. 1967).

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

$\begin{displaymath} 2^8+7^8+8^8+16^8+17^8+20^8+20^8+24^8+24^8=11^8+27^8 \end{displaymath}$

(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

$\begin{displaymath} 14^8+18^8+44^8+44^8+66^8+70^8+92^8+93^8+96^8+106^8+112^8=125^8 \end{displaymath}$

(9)

(Lander et al. 1967).

The smallest 12-1 solution is

$\begin{displaymath} 8^8+8^8+10^8+24^8+24^8+24^8+26^8+30^8+34^8+44^8+52^8+63^8=65^8 \end{displaymath}$

(10)

(Lander et al. 1967).

The general identity

$\begin{displaymath} (2^{8k+4}+1)^8=(2^{8k+4}-1)^8+(2^{7k+4})^8+(2^{k+1})^8+7[(2^{5k+3})^8+(2^{3k+2})^8] \end{displaymath}$

(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.