The primitive-divisor conjecture for the sequence $2^d-3$

For each positive integer $i$, consider the integer $2^i-3$. A primitive prime divisor of $2^d-3$ is a prime divisor that divides none of the earlier terms $2^i-3$ with $i<d$. Primitive-divisor conjecture. For every $d\ge 8$, there is a primitive prime divisor of $2^d-3$; equivalently, after checking the cases $d<8$, the Zsigmondy set of the sequence $(2^i-3)$ is exactly $\{1,2,7\}$. This conjecture is used only for a short conjectural proof of the nonexistence of the relevant subgroup; the paper also gives a complete longer proof, so the surrounding group-theoretic conclusion is established independently.