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It is believed that $a^3 + b^3 = c^3 + k$ has solutions in positive integers a, b and c except when k = 0 (part of FLT, and very non-trivial proof by Gauss if I'm right) or k = 4 (modulo 9) or k = 5 (modulo 9), which are both trivial. On the other hand, for many k, even quite small ones, it is very hard to find solutions, and solutions might never be found ...


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$$ \begin{align} 360^2+540^2 & \approx (648.99922958\ldots)^2 \\[10pt] 360^2+540^2 + 1^2 & = 649^2 \end{align} $$


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The following shows that for $n=3$ the ratio can be as close to $1$ as we like. For any positive integer $m$ let: $a = 3m^3 + 3m^2 + 2m + 1$ $b = 3m^3 + 3m^2 + 2m$ $c = 3m^2 + 2m + 1$ $d = m$ Then it is readily shown that $a^3 = b^3 + c^3 + d^3$. If we focus on the 'wrong solution' $a^3 = b^3 + c^3$, the ratio of LHS to RHS is: ...



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