Cube = Square - Square

Algebra Level 1

Calvin is trying to write a perfect cube as the difference of 2 perfect squares. He notices that

\[ \begin{array} { r r r r r r } 1^3 & = & 1^2 & - & 0^2 \\ 2^3 & = & 3^2 & - & 1^2 \\ 3^3 & = & 6^2 & - & 3 ^ 2 \\ 4^3 & = & 10^2 & - & 6^2 \\ \vdots & & & \vdots & \\ \end{array} \]

If he continues this pattern, he guesses that \( 7^3 = a^2 - b^2 \). What is \( a \times b \)?

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