Euler Quartic Conjecture

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

$$A^4+B^4=C^4+D^4.$$

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

See also Diophantine Equation--Quartic

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.