We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Let 
$d \ge 3$ be an integer and let 
$P \in \mathbb{Z}[x]$ be a polynomial of degree d whose Galois group is 
$S_d$. Let 
$(a_n)$ be a non-degenerate linearly recursive sequence of integers which has P as its characteristic polynomial. We prove, under the generalised Riemann hypothesis, that the lower density of the set of primes which divide at least one non-zero element of the sequence 
$(a_n)$ is positive.
Let F be a system of polynomial equations in one or more variables with integer coefficients. We show that there exists a univariate polynomial 
$D \in \mathbb {Z}[x]$ such that F is solvable modulo p if and only if the equation 
$D(x) \equiv 0 \pmod {p}$ has a solution.
