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Diophantine Equation--9th Powers

The 9.1.2 equation

A^9=B^9+C^9
(1)

is a special case of Fermat's last theorem with n=9, and so has no solution. No 9.1.3, 9.1.4, 9.1.5, 9.1.6, 9.1.7, 9.1.8, or 9.1.9 solutions are known. A 9.1.10 solution is

917^9=851^9+822^9+668^9+625^9+574^9+542^9+475^9+179^9+99^9+42^9
(2)

(J. Wroblewski 2002), and two 9.1.11 solutions are given by

252^9=247^9+202^9+167^9+133^9+108^9+87^9+74^9
(3)
+30^9+8^9+5^9+1^9
(4)
404^9=392^9+340^9+267^9+200^9+135^9+101^9+60^9
(5)
+52^9+44^9+9^9+4^9
(6)

(S. Chase; Aloril 2002). The smallest 9.1.12 solution is

103^9=91^9+91^9+89^9+71^9+68^9+65^9 +43^9+42^9+19^9+16^9+13^9+5^9
(7)

(Meyrignac 1997). No 9.1.13 solution is known. The smallest 9.1.14 solution is

66^9=63^9+54^9+51^9+49^9+38^9+35^9+29^9 +24^9+21^9+12^9+10^9+7^9+2^9+1^9
(8)

(Ekl 1998).

No 9.2.2, 9.2.3, 9.2.4,. 9.2.5, 9.2.6, 9.2.7, or 9.2.8 solutions are known. 9.2.9 solutions include

137^9+69^9=121^9+116^9+116^9+115^9+89^9
(9)
+52^9+28^9+26^9+14^9+9^9
(10)
686^9+429^9=661^9+589^9+484^9+326^9
(11)
+290^9+236^9+203^9+140^9+106^9
(12)

(J. Wroblewski 2002). A 9.2.10 solution is given by

121^9+2·116^9+115^9+89^9+52^9+28^9+26^9+14^9+9^9=137^9+69^9
(13)

(L. Morelli 1999). No 9.2.11 solutions are known. The smallest 9.2.12 solution is

4·2^9+2·3^9+4^9+7^9+16^9+17^9+2·19^9=15^9+21^9
(14)

(Lander et al. 1967, Ekl 1998). There are no known 9.2.13 or 9.2.14 solutions. The smallest 9.2.15 solution is

2^9+2^9+4^9+6^9+6^9+7^9+9^9+9^9+10^9+15^9 +18^9+21^9+21^9+23^9+23^9=26^9
(15)

(Lander et al. 1967).

There are no known 9.3.3, 9.3.4, 9.3.5, 9.3.6, 9.3.7, or 9.3.8 solutions. The smallest 9.3.9 solution is

2·38^9+3^9=41^9+23^9+2·20^9+18^9+2·13^9+12^9+9^9
(16)

(Ekl 1998). There is no known 9.3.10 solution. The smallest 9.3.11 solution is

2^9+3^9+6^9+7^9+9^9+9^9+19^9+19^9+21^9+25^9+29^9 =13^9+16^9+30^9
(17)

(Lander et al. 1967).

No 9.4.4 or 9.4.5 solutions are known. The smallest 9.4.6 solution is

90^9+64^9+35^9+35^9=86^9+80^9+62^9+43^9+27^9+16^9.
(18)

There are no known 9.4.7 or 9.4.8 solutions. The smallest 9.4.9 solution is

38^9+31^9+12^9+2^9 =36^9+2·32^9+30^9+15^9+13^9+8^9+4^9+3^9
(19)

(Ekl 1998). The smallest 9.4.10 solutions are

2^9+6^9+6^9+9^9+10^9+11^9+14^9+18^9+19^9+19^9 =5^9+12^9+16^9+21^9
(20)

(Lander et al. 1967).

The smallest 9.5.5 solution is

192^9+101^9+91^9+30^9+26^9 =180^9+175^9+116^9+17^9+12^9.
(21)

There is no known 9.5.6 solution. The smallest 9.5.7 solution is

35^9+26^9+2·15^9+12^9=33^9+32^9+24^9+16^9+14^9+8^9+6^9
(22)

(Ekl 1998). There are no known 9.5.8, 9.5.9, or 9.5.10 solutions. The smallest 9.5.11 solution is

3^9+5^9+5^9+9^9+9^9+12^9+15^9+15^9+16^9+21^9+21^9 =7^9+8^9+14^9+20^9+22^9
(23)

(Lander et al. 1967).

The smallest 9.6.6 solutions are

23^9+18^9+14^9+13^9+13^9+1^9=22^9+21^9+15^9+10^9+9^9+5^9
(24)
31^9+23^9+21^9+14^9+9^9+2^9=29^9+29^9+15^9+11^9+10^9+6^9
(25)
46^9+44^9+27^9+27^9+27^9+9^9=48^9+39^9+23^9+15^9+13^9+12^9
(26)
47^9+47^9+22^9+22^9+12^9+4^9=50^9+39^9+35^9+13^9+10^9+7^9
(27)
54^9+52^9+48^9+47^9+46^9+14^9=60^9+18^9+17^9+16^9+15^9+15^9
(28)
70^9+44^9+36^9+33^9+19^9+4^9=64^9+63^9+57^9+47^9+22^9+13^9
(29)
68^9+58^9+50^9+46^9+41^9+7^9=70^9+48^9+26^9+25^9+23^9+18^9
(30)

(Lander et al. 1967, Ekl 1998).

Ekl (1998) mentions but does not list nine primitive solutions to the 9.7.7 equation.

Moessner (1947) gives a parametric solution to the 9.10.10 equation.

Palamá (1953) gave a solution to the 9.11.11 equation.

Moessner and Gloden (1944) give the 9.11.12 solution

72^9+67^9+66^9+53^9+43^9+37^9+35^9+29^9+19^9+6^9+5^9 =71^9+70^9+63^9+55^9+40^9+39^9+33^9+32^9+17^9+9^9+2^9+1^9.
(31)

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References

Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309-1315, 1998.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. "On Equal Sums of Like Powers." Math. Student 15, 83-88, 1947.Moessner, A. and Gloden, A. "Einige Zahlentheoretische Untersuchungen und Resultate." Bull. Sci. École Polytech. de Timisoara 11, 196-219, 1944.Palamá, G. "Diophantine Systems of the Type sum_(i=1)^(p)a_i^k=sum_(i=1)^(p)b_i^k (k=1, 2, ..., n, n+2, n+4, ..., n+2r)." Scripta Math. 19, 132-134, 1953.

Referenced on Wolfram|Alpha

Diophantine Equation--9th Powers

Cite this as:

Weisstein, Eric W. "Diophantine Equation--9th Powers." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DiophantineEquation9thPowers.html

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