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Old 2020-03-22, 08:39   #6
R. Gerbicz
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"Robert Gerbicz"
Oct 2005

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Originally Posted by CRGreathouse View Post
It seems rather difficult to find positive integers n such that there are no positive integers a, b, and c with 10^n = a^3 + b^3 + c^2. n = 5 is an example, but there are no others up to 18.

I don't know of any modular obstructions.
Multiple the equation by 10^(6*k), since it is cube and square if there is a solution for n=N, then there is a solution for N+6*k.
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