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

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Answer 1 · 2017-11-29T16:27:52

See below.

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.