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PEN Math problems from the collection "Problems in Elementary Number Theory"

Math problems from the collection "Problems in Elementary Number Theory"

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PEN Math problems from the collection "Problems in Elementary Number Theory"

Math problems from the collection "Problems in Elementary Number Theory"

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Peter

#1 h May 25, 2007, 12:24 AM • 2 Y

Y by Adventure10, Mango247

Let $x, y, z$ non-zero real numbers such that $xy$ , $yz$ , $zx$ are rational.

Show that the number $x^{2}+y^{2}+z^{2}$ is rational.
If the number $x^{3}+y^{3}+z^{3}$ is also rational, show that $x$ , $y$ , $z$ are rational.

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edriv

#2 h May 25, 2007, 12:24 AM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Part a) is trivial:
$x^2 = \frac{xy \cdot zx}{yz}$

Then $x^2,y^2,z^2$ are rational numbers.

Let $k = x^3+y^3+z^3 \in \mathbb{Q}$ .
Then $(k-x^3)^2 = (y^3+z^3)^2 = (y^2)^3 + (z^2)^3 + 2(yz)^3 \in \mathbb{Q}$
$k^2 + x^6 -2kx^3 \in \mathbb{Q}$ , but $k, x^2 \in \mathbb{Q}$ .
Therefore $2kx^3 \in \mathbb{Q} \Rightarrow x^3 \in \mathbb{Q} \Rightarrow x = \frac{x^3}{x^2} \in \mathbb{Q}$ .

In the same way we prove that $y,z \in \mathbb{Q}$ .

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#3 h Feb 21, 2009, 6:59 PM • 2 Y

Y by Adventure10, Mango247

Hi,

Edriv

what about the case where k=0?
in fact this cannot occur because of Fermat's last Theorem (n=3) and the fact that x/z and y/z are both rational numbers.

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MBGO

#4 h Jan 16, 2013, 8:40 PM • 2 Y

Y by Adventure10, Mango247

$\frac{x}{y}$ (and same about other pairs) is rational, so as we can represent an irrartional number by $Aa$ such that $A$ is rational and $a$ is irrational,which it cannot devide into productions of an irrational and a rational, so let $x=Aa,y=Bb,Z=Cc;$ by what we have at the first we will get that $a=b=c$ ,the rest is obviouse.

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#5 h May 28, 2014, 9:17 PM • 1 Y

Y by Adventure10

From part a) it follows that the numbers $x^2, y^2, z^2,\frac{x}{ y}, \frac{y}{ z}, \frac{z}{ x}$ are rational, and since

$\frac{x^3+y^3+z^3}{y}=x^2(\frac{x}{ y})+y^2+z^2(\frac{z}{y})$ ,

$y$ is rational and so are also $x, z$ .

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