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Diophantine Equation--7th Powers


The 7.1.2 equation

 A^7+B^7=C^7
(1)

is a special case of Fermat's last theorem with n=7, 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,

 568^7=525^7+439^7+430^7+413^7+266^7+258^7+127^7
(2)

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

 12^7+35^7+53^7+58^7+64^7+83^7+85^7+90^7=102^7
(3)

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

 6^7+14^7+20^7+22^7+27^7+33^7+41^7+50^7+59^7=62^7
(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

 125^7+24^7=121^7+94^7+83^7+61^7+57^7+27^7
(5)

(Meyrignac). The smallest 7.2.8 solution is

 5^7+6^7+7^7+15^7+15^7+20^7+28^7+31^7=10^7+33^7
(6)

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

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

96^7+41^7+17^7=87^7+2·77^7+68^7+56^7
(9)
153^7+43^7+14^7=140^7+137^7+59^7+42^7+42^7.
(10)

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

 7^7+7^7+12^7+16^7+27^7+28^7+31^7=26^7+30^7+30^7
(11)

(Lander et al. 1967).

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

149^7+123^7+14^7+10^7=146^7+129^7+90^7+15^7
(12)
194^7+150^7+105^7+23^7=192^7+152^7+132^7+38^7
(13)
354^7+112^7+52^7+19^7=343^7+281^7+46^7+35^7
(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

50^7+43^7+16^7+12^7=52^7+29^7+26^7+11^7+3^7
(15)
81^7+58^7+19^7+9^7=77^7+68^7+56^7+48^7+2^7
(16)
87^7+74^7+69^7+40^7=82^7+79^7+75^7+25^7+9^7
(17)
99^7+76^7+32^7+29^7=93^7+88^7+68^7+36^7+35^7
(18)
98^7+82^7+58^7+34^7=99^7+75^7+69^7+16^7+13^7
(19)
104^7+96^7+60^7+14^7=102^7+95^7+81^7+57^7+23^7
(20)
111^7+102^7+40^7+29^7=112^7+96^7+82^7+55^7+21^7
(21)
113^7+102^7+86^7+23^7=120^7+81^7+58^7+55^7+10^7
(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

 8^7+8^7+13^7+16^7+19^7=2^7+12^7+15^7+17^7+18^7
(23)
 4^7+8^7+14^7+16^7+23^7=7^7+7^7+9^7+20^7+22^7
(24)
 11^7+12^7+18^7+21^7+26^7=9^7+10^7+22^7+23^7+24^7
(25)
 6^7+12^7+20^7+22^7+27^7=10^7+13^7+13^7+25^7+26^7
(26)
 3^7+13^7+17^7+24^7+38^7=14^7+26^7+32^7+32^7+33^7
(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

 2^7+3^7+6^7+6^7+10^7+13^7=1^7+1^7+7^7+7^7+12^7+12^7
(28)

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

 87^7+233^7+264^7+396^7+496^7+540^7 
 =90^7+206^7+309^7+366^7+522^7+523^7.
(29)

Moessner and Gloden (1944) gave the 7.9.10 solution

 42^7+37^7+36^7+29^7+23^7+19^7+13^7+6^7+5^7 
=41^7+40^7+33^7+28^7+27^7+15^7+14^7+9^7+2^7+1^7.
(30)

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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." http://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 x^7+y^7+z^7=0." §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.

Cite this as:

Weisstein, Eric W. "Diophantine Equation--7th Powers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DiophantineEquation7thPowers.html

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