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$$f(n) = n \ \text{xor} \ 1$$ Cheesy solution

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$a_0 = 0$, $a_{n+1} =1-a_n$.

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Simply $f(n) = n \space AND \space 1$ would work. (bitwise AND)

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In fact, there are algorithms for finding all integer solutions to any elliptic curve over the rationals. See, for example, the paper Solving elliptic diophantine equations: the general cubic case. In particular, the authors mention explicit upper bounds for integer solutions, so in principle we can find all integer solutions just by checking a finite ...

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Theorem by Baker (see here, page $20$): If $y^2=x^3+k$ for some $x,y,k\in\Bbb Z, k\neq 0$, then $$\max\{|x|,|y|\}\le \left(e^{10^{10}}\right)^{|k|^{10\, 000}}$$ Practically, it's not solvable for all $k\in\Bbb Z_{\neq 0}$ because of computer power/time constraints. Mordell's equation is fully solved when $0<|k|\le 10^4$ and for some \$10^4<|k|\le ...

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