Twice a square

Number Theory Level 5

Find a positive integer $N$ such that $p^3 + N^2$ is a perfect square for 2 distinct (positive) primes $p$ .

Details and assumptions

You may use a theorem of Ljunggren, which states that the equation $x^2+x+1=y^3$ has only the following integer solutions: $(x,y)=(0,1), (-1,1), (18,7), (-19,7)$ .