<s>[INST] For how many values of $a$ will $(a + 1)(a - 1)$ be prime? For how many values of $a$ will the number $(a+1)(a-1)$ be prime?
I showed that $a$ must be even because if it is odd then $(a-1)(a+1)$ will be even and therefore not prime. If it is even, then I also showed that $a$ must be a multiple of $3$ because otherwise $(a-1)(a+1)$ will be a product of $3$. Seems like it's not the right method to find the number of $a$ that will give $(a+1)(a-1)$ prime?
 [/INST] If you allow non integer values of $a$ then for any prime $p$ we have that when $a=\sqrt{p+1}$ that $(a-1)(a+1)=p$ so the answer is for infinitely many values of $a$. 
If you assume that $a$ is an integer then either $a-1=1$ and $a=2$ or $a-1=-1$ but then $a=0$ and $(a-1)(a+1)=-1$ not a prime number or $a+1=1$ but then $a=0$ again and if $a+1=-1$ then $a=-2$ which gives $(a-1)(a+1)=3$ then the answer is two values $\{2,-2\}$. 
If you assume that $a$ is a positive integer then the solution is one value $\{2\}$.
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