Let $$E = \{p^4 + p^2 - 2 \mid p \text{ is a prime},\; p > 3\}$$. What is the largest positive integer that divides all the numbers in $$E$$?

Factor the expression as $$p^4 + p^2 - 2 = (p^2 - 1)(p^2 + 2)$$. For a prime $$p > 3$$ the numbers $$p - 1$$ and $$p + 1$$ are consecutive even numbers with one divisible by 4, so $$8 \mid p^2 - 1$$; also $$p^2 \equiv 1 \pmod 3$$, so both $$p^2 - 1$$ and $$p^2 + 2$$ are divisible by 3 and their product by 9. Hence every member is divisible by $$8 \times 9 = 72$$, and since $$p = 5$$ and $$p = 7$$ give $$648 = 72 \times 9$$ and $$2448 = 72 \times 34$$ with $$\gcd(9, 34) = 1$$, nothing larger works.