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If $n$ is a positive integer, let $\phi(n)$ the Euler function.

( if $n=p_1^{\alpha_1}\dots p_k^{\alpha_k}$ with $p_i$ distinct primes, we have $\phi(n)=p_1^{\alpha_1-1} \dots p_k^{\alpha_k-1}(p_1-1)\dots(p_k-1)$ )

Let $P$ a polynomial in $\mathbb{Z}[X,Y]$.

We suppose there exists an infinite number of positive integer $n$ such that $P(n,\phi(n))=0$

Is $P$ reducible in factors of degree one ?

Thanks in advance.

1 Answers 1

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If $p(x,y)=(x-y)^2-x$, then $p(q^2,\phi(q^2))=0$ for all primes $q$.

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    k thanks for the link.2011-12-19