Find the positive integer solutions to the equation x^3-y^3=x y+61?

1 Answer
Jul 1, 2017

See below.

Explanation:

With p= 61

Making the transformation

{(x+y=a),(x-y=b):}

equivalent to a rotation of pi/4 clockwise with the purpose of cross product xy elimination, we obtain

b^2 + b^3 + a^2 (3 b-1) - 4 p=0 then

a^2=(4p-b^3-b^2)/(3b-1)

Here a^2 > 0 and p=61 then

b lt 6

Now evaluating (4*61-b^3-b^2)/(3b-1) for b in {1,2,cdots,5} we have

{121, 232/5, 26, 164/11, 47/7}

The only feasible outcome is 121=11^2 for b=1

so we have

{(x+y=11),(x-y=1):}

giving

x=6 and y = 5