Suppose that x , y and z are three integers (positive,negative or zero) such that w

Question

Suppose that

x

,

y

and

z

are three integers (positive,negative or zero) such that we get the following relationships simultaneously
1.

x + y = 1 - z

and
2.

x^{3} + y^{3} = 1 - z^{2}

Find all such

x

,

y

and

z

.

Answer 1

If

z = 1

, then

y = - x

and all triples

(x, - x, 1)

are solutions. So let's suppose

z \neq 1

. Then we can divide

\begin{matrix} (3) & x^{2} - x y + y^{2} = \frac{x^{3} + y^{3}}{x + y} = \frac{1 - z^{2}}{1 - z} = 1 + z . \end{matrix}

Adding (

(3)

to

(1)

yields

x^{2} - x y + y^{2} + x + y = 2.

Multiplying by

2

and adding

2

to both sides yields

\begin{aligned} 6 & = 2 x^{2} - 2 x y + 2 y^{2} + 2 x + 2 y + 2 \\ = (x^{2} - 2 x y + y^{2}) + (x^{2} + 2 x + 1) + (y^{2} + 2 y + 1) \\ = (x - y)^{2} + (x + 1)^{2} + (y + 1)^{2} . \end{aligned}

There aren't many combinations of

x

and

y

left to check.

Answers (1)