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.