In 1986, Noam Elkies found a method to construct an infinite series of counterexamples for the $k = 4$ case. His smallest counterexample was $2682440^4 + 15365639^4 + 18796760^4 = 20615673^4$. A particular case of Elkies' solutions can be reduced to the identity. $(85v^2 + 484v − 313)^4 + (68v^2 − 586v + 10)^4 + (2u)^4 = (357v^2 − 204v + 363)^4$ where $u^2 = 22030 + 28849v − 56158v^2 + 36941v^3 − 31790v^4$. This is an elliptic curve with a rational point at $v1 = −31/467$. From this initial rational point, one can compute an infinite collection of others. Substituting $v1$ into the identity and removing common factors gives the numerical example cited above. This paper is one of the shortest papers ever published by a serious Mathematical Journal. L.J. Lander and T.R. Parkin show a counterexample of Euler's sum of powers conjecture. Euler conjectured in 1769 that at least $n$ $n^{th}$ powers are required for $n>2$ to provide a sum that is itself an $n^{th}$ power. In other words if $\sum_{i=1}^{n}a_i^k = b^k$ where $n>2$ and $a_1, a_2, \dots, a_n, b$ are non-zero integers, then $n\geq k$. The conjecture represents an attempt to generalize Fermat's last theorem, which is the special case $n = 2$: if $a_1^k + a_2^k = b^k$, then $2 \geq k$. In 1966 Lander and Parkin through a direct computer search on a CDC 6600, found a counterexample for k = 5.  There's a detailed discussion [here](http://www.ams.org/journals/mcom/1967-21-099/S0025-5718-1967-0222008-0/S0025-5718-1967-0222008-0.pdf) of how the authors approached the problem from a computational standpoint.