find-all-pairs-of-integers-x-y-such-that-x-3-y-3-x-y-2-

Question Number 115906 by Eric002 last updated on 29/Sep/20

f i n d a l l p a i r s o f i n t e g e r s (x, y) s u c h t h a t

x^{3} + y^{3} = {(x + y)}^{2}

Answered by mindispower last updated on 29/Sep/20

x^{3} + y^{3} = (x + y) (x^{2} + y^{2} - x y)

\Leftrightarrow (x + y) (x^{2} + y^{2} - x y - (x + y)) = 0

x = - y

o r

x^{2} + y^{2} - x y - (x + y) = 0

y^{2} + y (- 1 - x) + x^{2} - x = 0

Δ = x^{2} + 2 x + 1 - 4 x^{2} + 4 x ⩾ 0

\Rightarrow 6 x - 4 x^{2} + 1 ⩾ 0

x \in [\frac{- 6 + \sqrt{48}}{- 8}, \frac{6 + \sqrt{48}}{8}]

0 ⩽ x ⩽ 1

x = 0 \Rightarrow y^{2} - y = 0 \Rightarrow y \in {0, 1}

x = 1 \Rightarrow y^{2} - 2 y = 0

y \in {0, 2}

S = {(a, - a); (0, 1); (0, 2)}