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by Ciro Santilli (@cirosantilli, 37)

Waring problem with negative numbers allowed

 ... Diophantine equation Additive number theory Additive basis Additive basis theorem Waring's problem Waring problem variant
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    • Sum of three cubes Waring problem with negative numbers allowed

Sum of three cubes (Every number has infinitely many representations as the sum of three cubes)

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Waring problem with negative numbers allowed Tags: Open problem in mathematics
Compared to Waring's problem, this is potentially much harder, as we can go infinitely negative in our attempts, there isn't a bound on how many tries we can have for each number.
In other words, it is unlikely to have a Conjecture reduction to a halting problem.
Video 1.
3 as the sum of the 3 cubes by Numberphile (2019)
Source.

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