Square Pyramidal Primes

We can easily see that $P_2 = 5$ is prime. Now we need to prove that there are no larger square pyramidal numbers which are prime.

The closed form expression for square pyramidal numbers is $P_n = \dfrac{n(n+1)(2n+1)}{6}$ .

$P_3 = 14$ is not a prime. Now if $n > 3$ , $n+1 > 4$ and $2n+1 > 7$ .

Now if we divide $n(n+1)(2n+1)$ by $6$ we are left with a product of at least $2$ integers which are both greater than $1$ because each of $n, n+1, 2n+1$ are greater than any of $2, 3$ .

Since a product of two integers greater than $1$ cannot be a prime. There cannot be a larger square pyramidal number which is prime.

Hence the answer is $\boxed{5}$ .