Diophantine Equation--6th Powers
المؤلف:
Ekl, R. L.
المصدر:
"Equal Sums of Four Seventh Powers." Math. Comput. 65
الجزء والصفحة:
...
22-5-2020
2170
Diophantine Equation--6th Powers
The 6.1.2 equation
 |
(1)
|
is a special case of Fermat's last theorem with 
, and so has no solution. No 6.1.
solutions are known for 
(Lander et al. 1967; Guy 1994, p. 140). The smallest 6.1.7 solution is
 |
(2)
|
(Lander et al. 1967; Ekl 1998). The smallest primitive 6.1.8 solutions are
 |
(3)
|
(Lander et al. 1967). The smallest 6.1.9 solution is
 |
(4)
|
(Lander et al. 1967). The smallest 6.1.10 solution is
 |
(5)
|
(Lander et al. 1967). The smallest 6.1.11 solution is
 |
(6)
|
(Lander et al. 1967). There is also at least one 6.1.16 identity,
 |
(7)
|
(Martin 1893). Moessner (1959) gave solutions for 6.1.16, 6.1.18, 6.1.20, and 6.1.23 equations.
Ekl (1996) has searched and found no solutions to the 6.2.2
 |
(8)
|
with sums less than 
. No solutions are known to the 6.2.3 or 6.2.4 equations. The smallest primitive 6.2.5 equations are
(E. Brisse 1999, Resta 1999, Resta and Meyrignac 2003, Meyrignac). The smallest 6.2.6 equation is
 |
(14)
|
(Ekl 1998). The smallest 6.2.7 solution is
 |
(15)
|
(Lander et al. 1967). The smallest 6.2.8 solution is
 |
(16)
|
(Lander et al. 1967). The smallest 6.2.9 solution is
 |
(17)
|
(Lander et al. 1967). The smallest 6.2.10 solution is
 |
(18)
|
(Lander et al. 1967).
Parametric solutions are known for the 6.3.3 equation
 |
(19)
|
(Guy 1994, pp. 140 and 142). Known solutions are
(Rao 1934, Lander et al. 1967, Ekl 1998). Ekl (1998) mentions but does not list the 87 smallest solutions to the 6.2.6 equation. The smallest primitive 6.3.4 solutions are
(Lander et al. 1967, Ekl 1998).
Moessner (1947) gave three parametric solutions to the 6.4.4 equation. The smallest 6.4.4 solution is
 |
(58)
|
(Rao 1934, Lander et al. 1967). The smallest 6.4.4.4 solution is
 |
(59)
|
(Lander et al. 1967).
Moessner and Gloden (1944) give the 6.7.8 solution
 |
(60)
|
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.
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.
Martin, A. "On Powers of Numbers Whose Sum is the Same Power of Some Number." Quart. J. Math. 26, 225-227, 1893.
Meyrignac, J.-C. "Computing Minimal Equal Sums of Like Powers." https://euler.free.fr.
Meyrignac, J.-C. "Description of Resta's Algorithm." https://euler.free.fr/how.htm.
Moessner, A. "On Equal Sums of Like Powers." Math. Student 15, 83-88, 1947.
Moessner, A. "Einige zahlentheoretische Untersuchungen und diophantische Probleme." Glasnik Mat.-Fiz. Astron. Drustvo Mat. Fiz. Hrvatske Ser. 2 14, 177-182, 1959.
Moessner, A. and Gloden, A. "Einige Zahlentheoretische Untersuchungen und Resultate." Bull. Sci. École Polytech. de Timisoara 11, 196-219, 1944.
Rao, S. K. "On Sums of Sixth Powers." J. London Math. Soc. 9, 172-173, 1934.
Resta, G. and Meyrignac, J.-C. "The Smallest Solutions to the Diophantine Equation 
." Math. Comput. 72, 1051-1054, 2003.
Resta, G. "New Results on Equal Sums of Sixth Powers." Instituto di Matematica Computazionale, Pisa, Italy. April 1999. https://www.chez.com/powersum/Tr-b4-08.zip
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