Linear-density conjecture for primitive divisors in quadratic polynomial sequences

Let $\beta$ be a nonzero integer and define the sequence $P=(P_n)$ by

$$P_n=n^2+\beta.$$

A term $P_n$ has a primitive divisor if it has a prime divisor that divides no earlier nonzero term of the sequence. Suppose that $-\beta$ is not a square. If $\rho_{\beta}(N)$ denotes the number of terms $P_n$ with $n<N$ that have primitive divisors, then the primitive-divisor density conjecture.

$$\rho_{\beta}(N)\sim cN$$

for some constant $c$ satisfying $0<c<1$. The preceding theorem establishes infinitely many terms without primitive divisors, while this conjecture predicts that the terms with primitive divisors nevertheless have a positive density strictly less than one.