I can, can you?

Number Theory Level 4

Find the sum of all integral values of $\kappa$ such that there exists positive integers $x$ , $y$ and $z$ satisfying $x^2+y^2+z^2=\kappa xyz$

3 0

\infty

1 solution

Sudhanshu Mishra
Jun 21, 2015

Let us first consider the case when $x^{2}=y^{2}=z^{2}$

Hence, $x=y=z$

Then, $3 x^{2}=k x^{3}$

As $x\neq0$ Therefore $\frac{3}{x}=k$

$\therefore x={1,3}$ $\therefore k={3,1}$

And we can easily prove by divisibility that there is no solution for $x\neq y\neq z$ $\therefore \sum k= 1+3 = 4$

what do you mean prove by divisibility? good solution otherwise

Silver Vice - 5 years, 2 months ago

How do you know that there are no solutions for $\kappa$ when $x^2\neq y^2\neq z^2$ ?

Abdur Rehman Zahid - 5 years, 1 month ago

"And we can easily prove by divisibility that there is no solution for distinct x,y,z."

Consider k=3 and (x,y,z)=(2,5,29)

Ananda Bhaduri - 1 year, 11 months ago