Let \(x\) and \(y\) be real numbers such that \(x+y, x^{2}+y^{2}, x^{3}+y^{3}\) and \(x^{4}+y^{4}\) are integers. If \(n\) is a natural number, prove that \(x^{n}+y^{n}\) is an integer.

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## Comments

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TopNewestIf \( x \) and \( y \) are irrational then there is no way that the initial integer conditions hold, so let us write \( x = \dfrac{a}{m} \) and \( y = \dfrac{b}{n} \) where \( gcd(a,m) = gcd(b,n) = 1 \). Let us assume that \( m \ne n \), ie, there is some prime factor in \( m \) which is not in \( n \). Then \( x + y = \dfrac{a}{m} + \dfrac{b}{n} = \dfrac{an+bm}{mn} \). Considering the numerator, \( an + bm \equiv an \pmod{m} \) but there is a prime factor in \( m \) which is not in \( n \) and \( gcd(a,m) = 1 \) so \( an \not\equiv 0 \pmod{m} \) ie. the numerator can't be divisible by \( m \) and so \( x + y \) is not an integer. Therefore our assumption was wrong so \( x \) and \( y \) have the same denominator. So we will rewrite them as \( x = \dfrac{a}{m} \) and \( y = \dfrac{b}{m} \) where again \( gcd(a,m) = gcd(b,m) = 1 \).

We have that \( x + y \) is an integer so \( a + b \equiv 0 \pmod{m} \) so \( a \equiv -b \pmod{m} \). Squaring both sides of the congruence gives \( a^2 \equiv b^2 \pmod{m} \). But we also have that \( x^2 + y^2 \) is an integer so \( a^2 + b^2 \equiv 0 \pmod{m} \) so \( a^2 \equiv -b^2 \pmod{m} \) (in fact \( a^2 + b^2 \equiv 0 \pmod{m^2} \) but that is less useful to us). Combining these congruences gives \( b^2 \equiv -b^2 \pmod{m} \) so \( 2b^2 \equiv 0 \pmod{m} \). But \( gcd(b,m) = 1 \), so either \( m = 1 \) or \( m = 2 \).

If \( m = 1 \) then the result is trivial (since \(x \) and \( y \) themselves are integers ). If \( m = 2 \) then \( a \) and \( b \) are odd, so \( a^2 + b^2 \equiv 2 \pmod{4} \) but \( x^2 + y^2 = \dfrac{a^2 + b^2}{4} \) so that means that \( x^2 + y^2 \) is not an integer.

So your initial set of conditions only hold if \( x \) and \( y \) are integers themselves, following which the result is obvious.

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Oh, in fact, I can produce a counterexample to your claim: \(x = \sqrt{2}, y = -\sqrt{2}\) satisfies the conditions of the problem but not your claim that \(x,y\) are integers.

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How exactly?

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Another statement which is not true is "Let us assume that \( m\neq n \),

i.e., there is some prime factor in \(m\) which is not in \(n\)." The IE part doesn't necessarily follow from the assumption. For example, take \(m = 2, n = 4 \). Note that with these values, we have \( an \equiv 0 \pmod{m} \), which contradicts your conclusion. This could be fixed though, with better bookkeeping of the variables.Log in to reply

I was considering WLOG ( m > n ) and so either a prime factor or prime exponent are different which gives the same result

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Very nice solution!

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The key step is to show that \( xy \) is an integer, and then use (strong) Induction on the algebraic identity

\[ x^{n+1} + y^{n+1} = (x+y) (x^n + y^n) - xy ( x^{n-1} + y^{n-1} ). \]

Note that you have to use the \( x^4 + y^4 \) condition, as just the first three are not strong enough. For example, using \( x = \frac{\sqrt{2}}{2}, y = - \frac{ \sqrt{2} } {2} \), gives us \( x+y = 0, x^2 + y^2 = 1, x^3 + y^3 = 0\) but \(x^4 + y^4 = \frac{1}{2} \) (and the higher even powers clearly do not yield integers).

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I haven't found how to get \(xy\) is an integer; I only got \(xy\) is half an integer. I'm still figuring out how to get past that.

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To get that \(xy\) is half an integer, you only used the first 2 conditions. Namely,

\[ 2xy = (x+y)^2 - (x^2 + y^2 ) \in \mathbb{N}. \]

Use the "hint" that you must use the 4th condition. Somehow or other, it must come into play (and it does, in a very similar manner).

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Hi Calvin

When I was doing the problem I also got stuck with xy. But can u further elaborate on the later part of your hint? Thanks

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This was my method!

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I would greatly appreciate help too.

I am serious.

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You are totally 44

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you are totally 18

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We use the identity \(x^n+y^n=(x+y)(x^{n-1}+y^{n-1})-xy(x^{n-2}+y^{n-2})\).

First we prove that \(xy\) is an integer. Note that \(2xy=(x+y)^2-(x^2+y^2)\in \mathbb{Z}\) and \(2x^2y^2=(x^2+y^2)^2-(x^4+y^4)\in \mathbb{Z}\). From the first equation we see that \(2xy=m\) for some integer \(m\), so \(xy=\frac{m}{2}\). From the second equation we have \(2x^2y^2=n\) for some integer \(n\). Plugging \(xy=\frac{m}{2}\) yields \(2\left(\frac{m^2}{4}\right)=n\Leftrightarrow \frac{m^2}{2}=n\in \mathbb{Z}\). So \(2\mid m\). Hence \(xy=\frac{m}{2}\in \mathbb{Z}\), as desired. //

Now, let \(p(n)=\text{true}\Rightarrow x^n+y^n\in \mathbb{Z}\). By the identity, \(p(1), p(n-1), xy, p(n-2)\Rightarrow p(n)\). So \(p(1), p(4), xy, p(3)\Rightarrow p(5)\); \(p(1), p(5), xy, p(4)\Rightarrow p(6)\), and by a straight-forward strong induction on \(n\), \(p(n)=\text{true}\ \forall n\in \mathbb{N}\). \(\Box\)

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STATE YOUR SOURCE

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