Question

Can you find all the nonzero natural integers x and y that satisfie the following relation ? (x+y)^3=(x-y-6)^2

Answer 1

See below.

Explanation:

Calling

`{((x+y)^3 = n),((x-y-6)^2 =m):}`

we have

`{(x+y = n^(1/3)),(x-y-6=m^(1/2)):}`

and then

`{(x = 1/2(n^(1/3)+m^(1/2)+6)),(y = 1/2(n^(1/3)-m^(1/2)-6)):}`

Now considering all the squares (`m`) and cubes (`n`) we will have an infinite lattice of pairs (`x,y`)

NOTE

Those non negative (`n. m`) should be both odd or both even.