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: \((x,y)=(0,1), (−1,1), (18,7), (−19,7)\).

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