## Diophantine Equation--5th Powers

The 2-1 fifth-order Diophantine equation

 (1)

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

are known, despite the fact that sums up to have been checked (Guy 1994, p. 140), improving on the results on Lander et al. (1967), who checked up to . (In fact, no solutions are known for Powers of 6 or 7 either.)

No solutions to the 3-1 equation

 (3)

are known (Lander et al. 1967), nor are any 3-2 solutions up to (Lander et al. 1967).

Parametric solutions are known for the 3-3 (Guy 1994, pp. 140 and 142). Swinnerton-Dyer (1952) gave two parametric solutions to the 3-3 equation but, forty years later, W. Gosper discovered that the second scheme has an unfixable bug. The smallest primitive 3-3 solutions are

 (4) (5) (6) (7) (8)

(Moessner 1939, Moessner 1948, Lander et al. 1967).

For 4 fifth Powers, we have the 4-1 equation

 (9)

(Lander and Parkin 1967, Lander et al. 1967), but it is not known if there is a parametric solution (Guy 1994, p. 140). Sastry's (1934) 5-1 solution gives some 4-2 solutions. The smallest primitive 4-2 solutions are
 (10) (11) (12) (13) (14) (15) (16) (17) (18) (19)

(Rao 1934, Moessner 1948, Lander et al. 1967).

A two-parameter solution to the 4-3 equation was given by Xeroudakes and Moessner (1958). Gloden (1949) also gave a parametric solution. The smallest solution is

 (20)

(Rao 1934, Lander et al. 1967). Several parametric solutions to the 4-4 equation were found by Xeroudakes and Moessner (1958). The smallest 4-4 solution is
 (21)

(Rao 1934, Lander et al. 1967). The first 4-4-4 equation is

 (22)

(Lander et al. 1967).

Sastry (1934) found a 2-parameter solution for 5-1 equations
 (23)
(quoted in Lander and Parkin 1967), and Lander and Parkin (1967) found the smallest numerical solutions. Lander et al. (1967) give a list of the smallest solutions, the first few being

 (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35)

(Lander and Parkin 1967, Lander et al. 1967).

The smallest primitive 5-2 solutions are

 (36) (37) (38) (39) (40) (41)

(Rao 1934, Lander et al. 1967).

The 6-1 equation has solutions

 (42) (43) (44) (45) (46) (47) (48) (49)

(Martin 1887, 1888, Lander and Parkin 1967, Lander et al. 1967).

The smallest 7-1 solution is

 (50)

(Lander et al. 1967).

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, p. 95, 1994.

Gloden, A. Über mehrgeradige Gleichungen.'' Arch. Math. 1, 482-483, 1949.

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. and Parkin, T. R. A Counterexample to Euler's Sum of Powers Conjecture.'' Math. Comput. 21, 101-103, 1967.

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. Methods of Finding th-Power Numbers Whose Sum is an th Power; With Examples.'' Bull. Philos. Soc. Washington 10, 107-110, 1887.

Martin, A. Smithsonian Misc. Coll. 33, 1888.

Martin, A. About Fifth-Power Numbers whose Sum is a Fifth Power.'' Math. Mag. 2, 201-208, 1896.

Moessner, A. Einige numerische Identitäten.'' Proc. Indian Acad. Sci. Sect. A 10, 296-306, 1939.

Moessner, A. Alcune richerche di teoria dei numeri e problemi diofantei.'' Bol. Soc. Mat. Mexicana 2, 36-39, 1948.

Rao, K. S. On Sums of Fifth Powers.'' J. London Math. Soc. 9, 170-171, 1934.

Sastry, S. On Sums of Powers.'' J. London Math. Soc. 9, 242-246, 1934.

Swinnerton-Dyer, H. P. F. A Solution of .'' Proc. Cambridge Phil. Soc. 48, 516-518, 1952.

Xeroudakes, G. and Moessner, A. On Equal Sums of Like Powers.'' Proc. Indian Acad. Sci. Sect. A 48, 245-255, 1958.