If $x$ and $y$ are numbers such that $x + y, x^2 + y^2, x^3 + y^3$ and $x^4 + y^4$ are all integers. Show that $x^n + y^n$ is an integer.

Question

If $x$ and $y$ are numbers such that $x + y, x^2 + y^2, x^3 + y^3$ and $x^4 + y^4$ are all integers. Show that $x^n + y^n$ is an integer for all positive integers $n$.

Using induction we have the base case for $n=1$ holding as $x+y$ is an integer by assumption. Assume now that the statement holds for $n$. Now $$(x+y)(x^n+y^n)=x^{n+1}+y^{n+1}+xy^n+yx^n.$$

The lhs is an integer so $x^{n+1}+y^{n+1}+xy^n+yx^n$ is also an integer. Now if $xy^n+yx^n$ is an intger can we conclude that $x^{n+1}+y^{n+1}$ is also an integer?

For $xy^n+yx^n$ we see that $$xy^n+yx^n=xy(y^{n-1}+x^{n-1})$$ and since $y^{n-1}+x^{n-1}$ is an integer we must show that $xy$ is an integer. Now $$(x+y)^2 = x^2 +y^2+2xy$$ and the lhs as well as $x^2+y^2$ are integers so $2xy$ is also an integer which makes $xy$ an integer.

Can we now conclude that $x^{n+1}+y^{n+1}$ is an integer?

How does knowing that $2xy\in \mathbb Z$ tell us that $xy\in \mathbb Z$? — lulu, Mar 16 '23 at 20:22
See https://artofproblemsolving.com/community/c6h220099p1220604 on AoPS. — Martin R, Mar 16 '23 at 20:27
Somewhat related: https://math.stackexchange.com/questions/820421/a-b-a2-b2-a3-b3-are-integers-implies-a-b-are-integers — Gerry Myerson, Mar 16 '23 at 22:09

score 1 · Answer 1 · answered Mar 16 '23 at 21:51

Note that indeed, $2xy$ is an integer. However, note also by the same reasoning $(x^4+y^4)=(x^2+y^2)^2-2x^2y^2$ that $2x^2y^2 = \frac{(2xy)^2}{2}$ is also an integer. This implies that $(2xy)^2$ cannot be an odd integer and thus $2xy$ cannot be an odd integer either. So $2xy$ must be an even integer, giving $xy$ an integer, and hence the result.

If $x$ and $y$ are numbers such that $x + y, x^2 + y^2, x^3 + y^3$ and $x^4 + y^4$ are all integers. Show that $x^n + y^n$ is an integer.

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