is a special case of Fermat's last theorem and so has no solutions for . Lander et al. (1967) give a table showing the
smallest
for which a solution to
(2)
with
is known. An updated table is given below; a more extensive table may be found at
Meyrignac's web site.
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, p. 101, 1994.Berndt,
B. C. and Bhargava, S. "Ramanujan--For Lowbrows." Amer. Math. Monthly100,
644-656, 1993.Dickson, L. E. History
of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover,
pp. 653-657, 2005.Gloden, A. Mehrgradige Gleichungen. Groningen,
Netherlands: P. Noordhoff, 1944.Guy, R. K. Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 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.Reznick,
B. Sums
of Even Powers of Real Linear Forms. Providence, RI: Amer. Math. Soc., 1992.Sekigawa,
H. and Koyama, K. "Nonexistence Conditions of a Solution for the Congruence
."
Math. Comput.68, 1283-1297, 1999.