How to Find the all integer solutions for: x + y + z = 3 x 3 + y 3 + z 3 = 3

Question

How to Find the all integer solutions for:  x  +  y  +  z  =  3      x    3    +      y    3    +      z    3    =  3

Savanah Hernandez · Accepted Answer

We have       x    3    +      y    3    =  3  −      z    3   and   x  +  y  =  3  −  z. Since   x  +  y divides       x    3    +      y    3  , we conclude that   z  −  3 divides       z    3    −  3, and therefore   z  −  3 divides   24. Similar considerations apply to   x and   y.So we are down to a finite and indeed fairly short list of candidates. We can use further little tricks to winnow the list.Remark: Let&#039;s throw in some number theory. It is a sometimes useful fact that       a    3   is always congruent to   0,   1, or   −  1 modulo   9. Thus if       x    3    +      y    3    +      z    3    =  3, we must have       x    3  ,       y    3  , and       z    3   all congruent to   1 modulo   9. It follows that all of   x  ,  y, and   z are congruent to   1 modulo   3, and hence so are   x  −  3,   y  −  3, and   z  −  3. The only divisors of   24 that satisfy this condition are   1  ,  4  ,  −  2  , and   −  8. So our only candidates for   x  ,  y  , and z are   4  ,  7  ,  1  , and   −  5.

Llubanipo · Accepted Answer

I won&#039;t give you the solution, but a path. First find all solutions where all numbers are &amp;gt;= 0, that&#039;s very easy. In other cases, you have both positive and negative numbers. Show why you can&#039;t have x = -y, for example. And then you show that the largest number cannot be very large.

How to Find the all integer solutions for: x + y + z = 3 x 3 </m

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