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Assuming I have functions $f, g : \mathbb{N} \to \mathbb{N}$ and I know that $g \circ f$ is a primitive recursive function. What can I tell about $f$ and $g$? Are they primitive recursive as well? Or at least one of them?

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  • $\begingroup$ What have you tried and where did you get stuck? Hint: you don't know anything. Find examples. $\endgroup$
    – Raphael
    Commented Jan 28, 2015 at 11:50
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    $\begingroup$ Hint: $g(x) = 1$ $\endgroup$
    – Pseudonym
    Commented Jan 28, 2015 at 13:21

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Let $f$ be a bijective function, which is not primitive recursive. We know, that such a function exists. Let further be $g=f^{-1}$ the inverse function of $f$. Therefore $f\circ g$ is the identity function, which is clearly primitive recursive.

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    $\begingroup$ Another fun example is if one function is constant, the other can be arbitrary (i.e. non-primitive recursive, or even noncomputable) $\endgroup$
    – Mike B.
    Commented Jan 28, 2015 at 15:17

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