## Diophantine Equation--6th Powers

The 2-1 equation

 (1)

is a special case of Fermat's Last Theorem with , and so has no solution. Ekl (1996) has searched and found no solutions to the 2-2
 (2)

with sums less than .

No solutions are known to the 3-1 or 3-2 equations. However, parametric solutions are known for the 3-3 equation

 (3)

(Guy 1994, pp. 140 and 142). Known solutions are
 (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

(Rao 1934, Lander et al. 1967).

No solutions are known to the 4-1 or 4-2 equations. The smallest primitive 4-3 solutions are

 (14) (15) (16) (17) (18)

(Lander et al. 1967). Moessner (1947) gave three parametric solutions to the 4-4 equation. The smallest 4-4 solution is
 (19)

(Rao 1934, Lander et al. 1967). The smallest 4-4-4 solution is

 (20)

(Lander et al. 1967).

No -1 solutions are known for (Lander et al. 1967). No solution to the 5-1 equation is known (Guy 1994, p. 140) or the 5-2 equation.

No solutions are known to the 6-1 or 6-2 equations.

The smallest 7-1 solution is

 (21)

(Lander et al. 1967). The smallest 7-2 solution is
 (22)

(Lander et al. 1967).

The smallest primitive 8-1 solutions are

 (23)

 (24)

 (25)

 (26)

 (27)

 (28)

 (29)

 (30)

 (31)

 (32)

 (33)

 (34)

 (35)

 (36)

(Lander et al. 1967). The smallest 8-2 solution is
 (37)

(Lander et al. 1967).

The smallest 9-1 solution is

 (38)

(Lander et al. 1967). The smallest 9-2 solution is
 (39)

(Lander et al. 1967).

The smallest 10-1 solution is

 (40)

(Lander et al. 1967). The smallest 10-2 solution is
 (41)

(Lander et al. 1967).

The smallest 11-1 solution is

 (42)

(Lander et al. 1967).

There is also at least one 16-1 identity,
 (43)
(Martin 1893). Moessner (1959) gave solutions for 16-1, 18-1, 20-1, and 23-1.

References

Ekl, R. L. Equal Sums of Four Seventh Powers.'' Math. Comput. 65, 1755-1756, 1996.

Guy, R. K. Sums of Like Powers. Euler's Conjecture.'' §D1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139-144, 1994.

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

Martin, A. On Powers of Numbers Whose Sum is the Same Power of Some Number.'' Quart. J. Math. 26, 225-227, 1893.

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

Moessner, A. Einige zahlentheoretische Untersuchungen und diophantische Probleme.'' Glasnik Mat.-Fiz. Astron. Drustvo Mat. Fiz. Hrvatske Ser. 2 14, 177-182, 1959.

Rao, S. K. On Sums of Sixth Powers.'' J. London Math. Soc. 9, 172-173, 1934.