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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.

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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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  • $\begingroup$ Gerry Myerson:Yes,thanks. $\endgroup$ – francis-jamet Dec 14 '11 at 15:50
  • $\begingroup$ @ Gerry: Do there exist examples of polynomials of above mentioned type which are not identically zero but have infinitely solutions? $\endgroup$ – nb1 Dec 16 '11 at 10:40
  • $\begingroup$ @Nikhil, I don't know. $\phi(n)$ is (in some sense) generally between $n^{1-\epsilon}$ and $n/\log\log n$ which makes it difficult for polynomial relations to hold without holding identically, but I don't see offhand how to turn this observation into a proof of anything. $\endgroup$ – Gerry Myerson Dec 17 '11 at 20:51
  • $\begingroup$ @GerryMyerson thanks for the clarification. Can you please give a link that describes the limits of $\phi(n)$ as given by you? $\endgroup$ – nb1 Dec 17 '11 at 21:03
  • $\begingroup$ @Nikhil, just type Euler phi into Google and go to the wikipedia article on the Euler totient function. $\endgroup$ – Gerry Myerson Dec 18 '11 at 16:31

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