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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 $n=7$, 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.



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© 1996-9 Eric W. Weisstein
1999-05-24