Diophantine Equation--7th Powers

The 7.1.2 equation

(1)

is a special case of Fermat's last theorem with , and so has no solution. No solutions to the 7.1.3, 7.1.4, 7.1.5, 7.1.6 equations are known. There is now a known solutions to the 7.1.7 equation,

(2)

(M. Dodrill 1999, PowerSum), requiring an update by Guy (1994, p. 140). The smallest 7.1.8 solution is

(3)

(Lander et al. 1967, Ekl 1998). The smallest 7.1.9 solution is

(4)

(Lander et al. 1967).

No solutions to the 7.2.2, 7.2.3, 7.2.4, or 7.2.5 equations are known. The smallest 7.2.6 equation is

(5)

(Meyrignac). The smallest 7.2.8 solution is

(6)

(Lander et al. 1967, Ekl 1998). A 7.2.10.10 solution is

			(7)
			(8)

(Lander et al. 1967).

No solutions to the 7.3.3 equation are known (Ekl 1996), nor are any to 7.3.4. The smallest 7.3.5 equations are

			(9)
			(10)

No solutions are known to the 7.3.6 equation. The smallest 7.3.7 solution is

(11)

(Lander et al. 1967).

Guy (1994, p. 140) asked if a 7.4.4 equation exists. The following solution provide an affirmative answer

			(12)
			(13)
			(14)

(Ekl 1996, 1998; M. Lau 1999; PowerSum). Numerical solutions to the 7.4.5 equation are given by Gloden (1949). The smallest primitive 7.4.5 solutions are

			(15)
			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)

(Lander et al. 1967, Ekl 1998).

Gloden (1949) gives parametric solutions to the 7.5.5 equation. The first few 7.5.5 solutions are

(23)

(24)

(25)

(26)

(27)

(Lander et al. 1967). Ekl (1998) mentions but does not list 107 primitive solutions to 7.5.5.

A parametric solution to the 7.6.6 equation was given by Sastry and Rai (1948). The smallest is

(28)

(Lander et al. 1967). Another found by Chen Shuwen is

(29)

Moessner and Gloden (1944) gave the 7.9.10 solution

(30)

REFERENCES:

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

Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309-1315, 1998.

Gloden, A. "Zwei Parameterlösungen einer mehrgeradigen Gleichung." Arch. Math. 1, 480-482, 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.; 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.

Moessner, A. and Gloden, A. "Einige Zahlentheoretische Untersuchungen und Resultate." Bull. Sci. École Polytech. de Timisoara 11, 196-219, 1944.

Nagell, T. "The Diophantine Equation ." §67 in Introduction to Number Theory. New York: Wiley, pp. 248-251, 1951.

Sastry, S. and Rai, T. "On Equal Sums of Like Powers." Math. Student 16, 18-19, 1948.