I was wondering - is there a way to determine which coefficient $a$ yields the most primes in this expression: $$a \cdot n^2 -1$$ where $n \in \mathbb{N}$ and it goes from $[\alpha , \beta] ~~ \alpha, \beta > 0, ~\beta > \alpha$
I tried to program a code which gets the range we want to check for $n$, $[0,1000]$ and the coefficients between $[1,1000]$ and got to the solution of $a=398$ which yields about $40.14\%$ primes.
Is there any way to determine this? It sounds interesting.
Thank you.