We make several improvements to methods for finding integer solutions to $x^3+y^3+z^3=k$ for small values of $k$. We implemented these improvements on Charity Engines global compute grid of 500,000 volunteer PCs and found new representations for several values of $k$, including $k=3$ and $k=42$. This completes the search begun by Miller and Woollett in 1954 and resolves a challenge posed by Mordell in 1953.
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