Square pyramidal numbers: P(n) = 1²+2²+...+n² = n(n+1)(2n+1)/6. First: 1,5,14,30,55,91,140,204,285... Count cannonballs in a square pyramid. The cannonball problem: only P(24)=4900=70² is both a square pyramidal and a perfect square.
How to Use
Enter n:
- P(n): The pyramidal number
- Sum: 1²+2²+...+n²
- Layers: Individual squares
The Cannonball Problem
When is a sum of consecutive squares also a perfect square? P(n) = m²? The only solution is n=24: P(24)=4900=70². Proved by G.N. Watson in 1918 using elliptic functions.
The Formula
P(n) = n(n+1)(2n+1)/6. Proof by induction or by the identity Σk² = n³/3 + n²/2 + n/6. This is a degree-3 polynomial in n, giving cubic growth.
Step-by-Step Instructions
- 1Enter n.
- 2Compute P(n).
- 3See individual squares.
- 4Check sequence.
- 5Compare to formula.