I have to proofe that there are functions defined by $f:\mathbb{N} \rightarrow \mathbb{N}, f(n)=f(2n), \forall n\in \mathbb{N}$, which are not-computable. However I'm not really sure about the correct method.

I thought about a proof by contradiction. Assume each of those functions are computable. Then, by the Church-turing-thesis, there has to exist a TM which can compute every of those functions. Therefore $L(M)=\{code(M) | \text{M calculates this type of function}\}$ would be decidable. However I profed earlier, that this language is undecidable. This would lead to a contradiction, but I'm not sure about the correctness of my profe...

Thank you for your help :)

  • $\begingroup$ The condition f(n) = f(2n) allows you to define f(n) any way you like for odd n, and then the values for even n are defined. That’s not much of a restriction. $\endgroup$ – gnasher729 Feb 13 '20 at 3:26

Cook up some encoding of Turing machines so that if $n$ codes a machine, $2 n$ codes "the same" (perhaps use binary numbers ending in 1 as starting points, and 0s at the end are disregarded, thus making that odd $n$ and $n \cdot 2^k$ represent the same machine). Then use that e.g. $\operatorname{HALT}(M)$ (does $M$ halt if started on an empty tape?) isn't computable.


Another way to do it: Take any non-computable function $g(n)$, then the function defined as:

$\begin{equation*} f(2^k (2 n + 1)) = g(n) \end{equation*}$

for all $k \ge 0$ satisfies your conditions.

  • $\begingroup$ Also: $f(p_n) = g(n)$ where $p_i$ is the ith prime number. $\endgroup$ – Pseudonym Feb 13 '20 at 4:31

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