Diophantine Equation--9th Powers

The 9.1.2 equation

(1)

is a special case of Fermat's last theorem with , and so has no solution. No 9.1.3, 9.1.4, 9.1.5, 9.1.6, 9.1.7, 9.1.8, or 9.1.9 solutions are known. A 9.1.10 solution is

917^9=851^9+822^9+668^9+625^9+574^9+542^9+475^9+179^9+99^9+42^9

(2)

(J. Wroblewski 2002), and two 9.1.11 solutions are given by

	(3)
	(4)
	(5)
	(6)

(S. Chase; Aloril 2002). The smallest 9.1.12 solution is

(7)

(Meyrignac 1997). No 9.1.13 solution is known. The smallest 9.1.14 solution is

(8)

(Ekl 1998). The smallest 9.1.15 solution is

(9)

(Lander et al. 1967).

No 9.2.2, 9.2.3, 9.2.4, 9.2.5, 9.2.6, 9.2.7, or 9.2.8 solutions are known. A 9.2.9 solution is

(10)

(J. Wroblewski 2002). A 9.2.10 solution is given by

121^9+2·116^9+115^9+89^9+52^9+28^9+26^9+14^9+9^9=137^9+69^9

(11)

(L. Morelli 1999). No 9.2.11 solutions are known. The smallest 9.2.12 solution is

(12)

(Lander et al. 1967, Ekl 1998). There are no known 9.2.13 or 9.2.14 solutions.

There are no known 9.3.3, 9.3.4, 9.3.5, 9.3.6, 9.3.7, or 9.3.8 solutions. The smallest 9.3.9 solution is

(13)

(Ekl 1998). There is no known 9.3.10 solution. The smallest 9.3.11 solution is

(14)

(Lander et al. 1967).

No 9.4.4 or 9.4.5 solutions are known. The smallest 9.4.6 solution is

(15)

There are no known 9.4.7 or 9.4.8 solutions. The smallest 9.4.9 solution is

(16)

(Ekl 1998). The smallest 9.4.10 solutions are

(17)

(Lander et al. 1967).

The smallest 9.5.5 solution is

(18)

There is no known 9.5.6 solution. The smallest 9.5.7 solution is

35^9+26^9+2·15^9+12^9=33^9+32^9+24^9+16^9+14^9+8^9+6^9

(19)

(Ekl 1998). There are no known 9.5.8, 9.5.9, or 9.5.10 solutions. The smallest 9.5.11 solution is

(20)

(Lander et al. 1967).

The smallest 9.6.6 solutions are

			(21)
			(22)
			(23)
			(24)
			(25)
			(26)
			(27)

(Lander et al. 1967, Ekl 1998).

Ekl (1998) mentions but does not list nine primitive solutions to the 9.7.7 equation.

Moessner (1947) gives a parametric solution to the 9.10.10 equation.

Palamá (1953) gave a solution to the 9.11.11 equation.

Moessner and Gloden (1944) give the 9.11.12 solution

72^9+67^9+66^9+53^9+43^9+37^9+35^9+29^9+19^9+6^9+5^9
=71^9+70^9+63^9+55^9+40^9+39^9+33^9+32^9+17^9+9^9+2^9+1^9.

(28)

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References

Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309-1315, 1998.Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446-459, 1967.Meyrignac, J.-C. "Computing Minimal Equal Sums of Like Powers." http://euler.free.fr.Moessner, A. "On Equal Sums of Like Powers." Math. Student 15, 83-88, 1947.Moessner, A. and Gloden, A. "Einige Zahlentheoretische Untersuchungen und Resultate." Bull. Sci. École Polytech. de Timisoara 11, 196-219, 1944.Palamá, G. "Diophantine Systems of the Type