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Date: 9-11-2020
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The 6.1.2 equation
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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
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(Lander et al. 1967; Ekl 1998). The smallest primitive 6.1.8 solutions are
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(Lander et al. 1967). The smallest 6.1.9 solution is
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(Lander et al. 1967). The smallest 6.1.10 solution is
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(Lander et al. 1967). The smallest 6.1.11 solution is
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(Lander et al. 1967). There is also at least one 6.1.16 identity,
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(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
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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
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(E. Brisse 1999, Resta 1999, Resta and Meyrignac 2003, Meyrignac). The smallest 6.2.6 equation is
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(Ekl 1998). The smallest 6.2.7 solution is
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(Lander et al. 1967). The smallest 6.2.8 solution is
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(Lander et al. 1967). The smallest 6.2.9 solution is
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(Lander et al. 1967). The smallest 6.2.10 solution is
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(Lander et al. 1967).
Parametric solutions are known for the 6.3.3 equation
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(Guy 1994, pp. 140 and 142). Known solutions are
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(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
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(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
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(Rao 1934, Lander et al. 1967). The smallest 6.4.4.4 solution is
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(Lander et al. 1967).
Moessner and Gloden (1944) give the 6.7.8 solution
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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.
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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