Question 4
A set of positive integers is called *fragrant* if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the smallest possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$
\{P(a+1),P(a+2),\dots,P(a+b)\}
$$ is fragrant?
\{P(a+1),P(a+2),\dots,P(a+b)\}
$$ is fragrant?