Swinnerton-Dyer had a paper, I think from the 1950s, that has a parametric solution to A5+B5+C5=D5+E5+F5. ---Rich ---- Quoting Frank Stevenson <frankstevensonmobile@gmail.com>:
As a programming exercise I just finished a search for a solution A? + B? = C? + D? for numbers up to 4.25e37 ( 2 ^ 125 ) but did not find anything. This is perhaps not a surprise, since the density of possible solutions falls precipitously. I ran the search over a couple of days on 8 x Tesla V100 GPUs, and can give details if anyone is interested.
But my main point of curiosity now is what the current thinking about this open problem is: Do any taxicab numbers exist for powers >= 5 ? Is it even something that is tackled seriously ?
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