There is this really nice paper by J.P.Serre on the congruence subgroup property for $SL\_2$ for $S$-arithmetic groups (<https://www.jstor.org/stable/1970630>). If one looks at the proof of Proposition 3 there, Serre in fact proves the following result. 


Let $a,b \in {\mathbb N}$ be two co-prime integers, and $\phi$ be Euler's totient function. For each $x\in {\mathbb N}$ we may consider $\phi (ax+b)$. Now consider the g.c.d. of the infinite set of numbers  

$$N(a,b)= g.c.d. \{ \phi (ax+b): x=1,2,3,\cdots \}.$$ Now $N(a,b)$ seemingly depends on $a,b$ but it does not much: $N(a,b)$ divides $8$. 


The proof of this uses Dirichlet's theorem on infinitude of primes. 


If ${\mathbb Q}$ is replaced by a number field $K$, and $a,b$ are co-prime integers, define $\phi (ax+b)$ to be the number of units in the quotient ring $O\_K/(ax+b)$, then the analogous g.c.d. divides $2\mu \_K^2$ where $\mu \_K$ is the number of roots of unity in $K$. 



> 
> My question is : if I replace the linear polynomial $ax+b$ by any polynomial $P(x)=a\_0+ a\_1x+\cdots+ a\_nx^n$, with the numbers $a\_0,a\_1, \cdots, a\_n$ co-prime and $a\_n\neq 0$, then does the corresponding g.c.d. 
>  $$g.c.d \{\phi (P(x)):x=0,1,2,..\}$$ depend (i.e. is bounded by a constant dependent) only on the degree $n$ and not on the polynomial? 
> 
> 
> 


The question came up in a question on discrete groups, which could be resolved, but THIS question remained. I do not have any applications for this, but I thought it was interesting on its own. 


[Edit] I should have added the link <https://arxiv.org/abs/math/0409377>. 


[Edit] The following paper <https://arxiv.org/abs/1909.10808> answers this affirmatively (unconditionally for $n=2$ and modulo a well known conjecture in the general case). So the answer is Yes. 



g.c.d. and Euler's totient function
I have made some computations which seem to corroborate the OP's conjecture, namely that for any $n$ there exists a $N$, such that for every polynomial $P$ of degree $n$, with positive integral coefficients and content 1, the quantity $$g(P):= g.c.d(\phi(P(x)),x \geq 1)$$
divides $N$.


For $n=1$, as the OP says, one can take $N=8$ as proved by Serre.


For $n=2$, it seems that one can take $N=2^4 3^2 = 144$. It seems even more
that one cannot do better, because for $P(x)=16x^2+32x+17$, I get experimentally $g(P)=16$
(this must not be hard to prove but I haven't tried), and for $P(x)=27 x^2 + 9x+1$, I get $g(P)=18$. So $144 | N$. On the other hand I have need been able to find any $P$ such that $g(P)$ was not a divisor of $144$.


For $n=3$ or $n=4$, I have failed to find any $P$ with $g(P)\geq 2$. This suggests $N=2$ in these cases.


