Recall that a perfect square is the square of some integer. How many perfect squares less than 10,000 can be represented as the difference of two consecutive perfect cubes?

Guest Aug 2, 2021

#1**0 **

Note that...

$1-0 = 1$

$4-1 = 3$

$9-4 = 5$

$16-9 = 7$

$25-16 = 9$

$...$

Thus, every positive odd number can be written as the difference between two perfect squares. We consider squares in the range of $1 - 99$, since $100^2 = 10,000$. An even number squared will never be odd, and vice versa an odd number squares will always be even.

Thus, taking $\frac{99+1}{2} = \boxed{50}$, which is our answer.

xCorrosive Aug 2, 2021