# All polynomials with no natural roots and integer coefficients such that $\phi(n)|\phi(P(n))$

Let $P(x)$ be a polynomial with integer coefficients such that the equation $P(x)=0$ has no positive integer solutions. Find all polynomials $P(x)$ such that for all positive integers $n$ we have $\phi(n) \mid \phi(P(n))$. It is conjectured there are none (other than the trivial $P(x) = x^k Q(x)$).

NOTE: For $\phi(P(n))$ to be well-defined, it has been suggested that we require $P(n) > 0$ for all positive integers $n$.

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@Amir, better be long and clear. –  lhf Jun 3 '11 at 14:30
@DJC: What about $p(x)=x$. –  Eric Naslund Jun 3 '11 at 16:35
@Eric: The point of the problem is to find all such polynomials. I only ask for one. If indeed there are no such polynomials, then these two tasks are one and the same. By the words "Classifying all such polynomials" I assumed that there might have been some nontrivial polynomials that fit the bill (other than those of the form $P(n) = n^kQ(n)$ which are excluded by the question). –  JavaMan Jun 3 '11 at 18:25
Actually, this problem was posted here. The first part have been solved, but the second part (this problem) is still unsolved. –  Amir Hossein Jun 4 '11 at 7:26
Shouldn't we actually require that $P(x) > 0$ for positive integers $x$, so that $\phi(P(x))$ is defined? For example, $P(x) = x^2-200$ is never zero but is negative for some values of $x$. –  Srivatsan Aug 13 '11 at 3:35