Suppose that x, y and z are three integers (positive,negative or zero) such that we get the following relationships simultaneously1. x + y = 1 − zand2. x 3 + y 3 = 1 − z 2 Find all such x, y and z.

Question

Suppose that   x,   y and   z are three integers (positive,negative or zero) such that we get the following relationships simultaneously1.   x  +  y  =  1  −  zand2.       x    3    +      y    3    =  1  −      z    2  Find all such   x,   y and   z.

Norah Baxter · Accepted Answer

If   z  =  1, then   y  =  −  x and all triples   (  x  ,  −  x  ,  1  ) are solutions. So let&#039;s suppose   z  ≠  1. Then we can divide                    (3)                              x          2                −        x        y        +                  y          2                =                                            x              3                        +                          y              3                                            x            +            y                          =                              1            −                          z              2                                            1            −            z                          =        1        +        z        .            Adding (  (  3  ) to   (  1  ) yields      x    2    −  x  y  +      y    2    +  x  +  y  =  2.Multiplying by   2 and adding   2 to both sides yields                    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                .            There aren&#039;t many combinations of   x and   y left to check.

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

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