A square pyramidal number counts the points in a square-based pyramid with \(k\) levels: \(P_k = k(k+1)(2k+1)/6\). The first: 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506.
This is also the sum of the first \(k\) perfect squares: \(P_k = 1^2 + 2^2 + \cdots + k^2\). The closed form was known to Archimedes.
The cannonball problem (Lucas, 1875) asks: which square pyramidal numbers are also perfect squares? The only nontrivial answer is \(P_{24} = 4900 = 70^2\). Proven by G. N. Watson in 1918, the result connects to elliptic curves and Pell equations.