Euler Quartic Conjecture

Euler conjectured that there are no Positive Integer solutions to the quartic Diophantine Equation

This conjecture was disproved by N. D. Elkies in 1988, who found an infinite class of solutions.

References

Berndt, B. C. and Bhargava, S. Ramanujan--For Lowbrows.'' Amer. Math. Monthly 100, 644-656, 1993.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139-140, 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.

Ward, M. Euler's Problem on Sums of Three Fourth Powers.'' Duke Math. J. 15, 827-837, 1948.