interesting question

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If $x+y$ , $x^2+y^2$ , $x^3+y^3$ , and $x^4+y^4$ are all integers, and $x$ and $y$ are both real numbers, do $x$ and $y$ have to be integers as well?

AvenJohn Feb 20, 2022

Guest · Answer 1 · 2022-02-20T04:47:43

I think so.

Here is my reasoning: $x^2+y^2= (x+y)^2-2xy$ , since we know x+y is an integer, and x^2+y^2 is also an integer, then -2xy must be an integer.

maybe someone else can do this as well?

AvenJohn · Answer 2 · 2022-02-20T10:53:01

yes, but just because 2xy is an integer, it doesn't say anything about x, y, or xy.

thats where im stumped

Melody · Answer 3 · 2022-02-21T09:59:28

I have played with this for ages but I have not come up with anything helpful :((