Lander, Parkin, and Selfridge Conjecture: Five Three

Special case of the Lander–Parkin–Selfridge conjecture: there is no solution in positive integers to $x_1^5 + x_2^5 + x_3^5 = y^5.$ That is, for all $x_1, x_2, x_3, y \in \mathbb{N}$ with $x_1, x_2, x_3, y > 0$, $x_1^5 + x_2^5 + x_3^5 \neq y^5.$ This corresponds to the case $k = 5$, $n = 3$, $m = 1$ of the general conjecture, where $n + m = 4 < 5 = k$ would be required to yield a counterexample.