Diophantine Equation--10th Powers

The 10.1.2 equation

(1)

is a special case of Fermat's last theorem with , and so has no solution. No  solutions are known with . A 10.1.13 solution is

(2)

(S. Chase). The smallest 10.1.15 solution is

(3)

(J.-C. Meyrignac 1999). The smallest 10.1.22 solution is

(4)

(Ekl 1998). The smallest 10.1.23 solution is

(5)

(Lander et al. 1967).

10.2.12 solutions include

	(6)
	(7)

(V. Pliousnine 2000, N. Kuosa 2000). The smallest 10.2.13 solution is

(8)

The smallest 10.2.15 solution is

(9)

(Ekl 1998). The smallest 10.2.19 solution is

(10)

(Lander et al. 1967). A 10.3.11 solution is

(11)

(J. Wroblewski 2002). A 10.3.12 solution is

(12)

(T. Nolan 2000). The smallest 10.3.13 solution is

(13)

The smallest 10.3.14 solution is

(14)

(Ekl 1998). The smallest 10.3.24 solution is

(15)

(Lander et al. 1967).

A 10.4.9 solution is

(16)

(J. Wroblewski 2002). 10.4.10 solutions include

	(17)
	(18)
	(19)
	(20)
	(21)
	(22)
	(23)
	(24)
	(25)
	(26)
	(27)
	(28)
	(29)
	(30)
	(31)
	(32)
	(33)
	(34)
	(35)
	(36)
	(37)
	(38)
	(39)
	(40)
	(41)
	(42)
	(43)
	(44)
	(45)
	(46)
	(47)
	(48)

(J. Wroblewski 2002). A 10.4.11 solution is

(49)

(S. Chase). The 10.4.12 equation has solution

(50)

(E. Bainville 1999). The smallest 10.4.15 solution is

(51)

(Ekl 1998). The smallest 10.4.23 solution is

(52)

(Lander et al. 1967).

The smallest 10.5.16 solutions are

	(53)
	(54)

(Lander et al. 1967, Ekl 1998).

The smallest 10.6.6 solution is

(55)

The smallest 10.6.16 solution is

(56)

(Ekl 1998). The smallest 10.6.27 solution is

(57)

(Lander et al. 1967).

The smallest 10.7.7 solutions are

	(58)
	(59)
	(60)
	(61)

(Lander et al. 1967, Ekl 1998).

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." https://euler.free.fr.