Squares and perfect cubes
There are 2 positive integers, the difference of their squares is a perfect cube , and the difference of their cubes is a square. Find the smallest possible numbers.
If these numbers are and , enter your answer as .
The answer is 60.
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1 solution
(a – b)(a + b) = p^3. a – b = p, a + b = p^2. So a = (〖 p〗^2+p)/2, and b = (〖 p〗^2-p)/2 a^3 - b^3 = q^2. By substitution, 1/4 p^3(3〖 p〗^2+1) = q^2. So the smallest p=4, then a=10, and b=6.