I am writing a paper on the topological structure of the Golomb space (defined here) and arrived to the following question:

**Question 1.** Is it true that for a number $a\in\mathbb N$ the equation $x^2+x=a$ has an integer solution $x$ if and only if for any number $b\in\mathbb N$, coprime with $a$, the equation $x^2+x=a \mod b$ has a solution $x$ (i.e., a solution in the ring $\mathbb Z/b\mathbb Z$).

In fact, I need a more general fact.

**Question $2^n$.** Is it true that for any number $n\ge 0$ and any number $a\in\mathbb N$ the equation $(x^2+x)^{2^n}=a$ has an integer solution $x$ if and only if for any number $p\in\mathbb N$, coprime with $a$, the equation $(x^2+x)^{2^n}=a \mod p$ has a solution $x$ (i.e., a solution in the ring $\mathbb Z/p\mathbb Z$).

We can also ask a more general

**Problem.** For which monic polynomials $f\in\mathbb Z[x]$ the following local-to-global principle holds:

$(*)$: for every $b\in\mathbb N$ the equation $f(x)=a$ has a solution in $\mathbb Z$ if and only if for any $b\in\mathbb N$, relatively prime with $a$ the equation $f(x)=a \mod b$ has a solution in the ring $\mathbb Z/b\mathbb Z$?

**Remark.** I have edited the second question a little bit.