Diophantine Equation--7th Powers

The 2-1 equation

$\begin{displaymath} A^7+B^7=C^7 \end{displaymath}$

(1)

is a special case of Fermat's Last Theorem with

, and so has no solution. No solutions to the 2-2 equation

$\begin{displaymath} A^7+B^7=C^7+D^7 \end{displaymath}$

(2)

are known.

No solutions to the 3-1 or 3-2 equations are known, nor are solutions to the 3-3 equation

$\begin{displaymath} A^7+B^7+C^7=D^7+E^7+F^7 \end{displaymath}$

(3)

(Ekl 1996).

No 4-1, 4-2, or 4-3 solutions are known. Guy (1994, p. 140) asked if a 4-4 equation exists for 7th Powers. An affirmative answer was provided by (Ekl 1996),

$\begin{displaymath} 149^7 + 123^7 + 14^7 + 10^7 = 146^7 + 129^7 + 90^7 + 15^7 \end{displaymath}$

(4)

$\begin{displaymath} 194^7 + 150^7 +105^7 + 23^7 = 192^7 + 152^7 +132^7 + 38^7. \end{displaymath}$

(5)

A 4-5 solution is known.

No 5-1, 5-2, or 5-3 solutions are known. Numerical solutions to the 5-4 equation are given by Gloden (1948). The smallest 5-4 solution is

$\begin{displaymath} 3^7+11^7+26^7+29^7+52^7=12^7+16^7+43^7+50^7 \end{displaymath}$

(6)

(Lander et al. 1967). Gloden (1949) gives parametric solutions to the 5-5 equation. The first few 5-5 solutions are

$\begin{displaymath} 8^7+ 8^7+13^7+16^7+19^7 = 2^7+12^7+15^7+17^7+18^7 \end{displaymath}$

(7)

$\begin{displaymath} 4^7+ 8^7+14^7+16^7+23^7 = 7^7+ 7^7+ 9^7+20^7+22^7 \end{displaymath}$

(8)

$\begin{displaymath} 11^7+12^7+18^7+21^7+26^7 = 9^7+10^7+22^7+23^7+24^7 \end{displaymath}$

(9)

$\begin{displaymath} 6^7+12^7+20^7+22^7+27^7 =10^7+13^7+13^7+25^7+26^7 \end{displaymath}$

(10)

$\begin{displaymath} 3^7+13^7+17^7+24^7+38^7 =14^7+26^7+32^7+32^7+33^7 \end{displaymath}$

(11)

(Lander et al. 1967).

No 6-1, 6-2, or 6-3 solutions are known. A parametric solution to the 6-6 equation was given by Sastry and Rai (1948). The smallest is

$\begin{displaymath} 2^7+3^7+6^7+6^7+10^7+13^7=1^7+1^7+7^7+7^7+12^7+12^7 \end{displaymath}$

(12)

(Lander et al. 1967).

There are no known solutions to the 7-1 equation (Guy 1994, p. 140). A 2-10-10 solution is

$\displaystyle 2^7+27^7$	$\textstyle =$	$\displaystyle 4^7+8^7+13^7+14^7+14^7+16^7+18^7+22^7+23^7+23^7$
	$\textstyle =$	$\displaystyle 7^7+7^7+9^7+13^7+14^7+18^7+20^7+22^7+22^7+23^7$	(13)

(Lander et al. 1967). The smallest 7-3 solution is

$\begin{displaymath} 7^7+7^7+12^7+16^7+27^7+28^7+31^7=26^7+30^7+30^7 \end{displaymath}$

(14)

(Lander et al. 1967).

The smallest 8-1 solution is

$\begin{displaymath} 12^7+35^7+53^7+58^7+64^7+83^7+85^7+90^7=102^7 \end{displaymath}$

(15)

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

$\begin{displaymath} 5^7+6^7+7^7+15^7+15^7+20^7+28^7+31^7=10^7+33^7 \end{displaymath}$

(16)

(Lander et al. 1967).

The smallest 9-1 solution is

$\begin{displaymath} 6^7+14^7+20^7+22^7+27^7+33^7+41^7+50^7+59^7=62^7 \end{displaymath}$

(17)

(Lander et al. 1967).

References

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

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.

Sastry, S. and Rai, T. ``On Equal Sums of Like Powers.'' Math. Student 16, 18-19, 1948.