PEN-2007-5

[Tahmid Hasan]

Let $x,y$ be positive integers such that $xy \mid x^2+y^2+1$.Show that $\frac {x^2+y^2+1}{xy}=3$.
Hint:

ever heard heard of root flipping?

[zadid xcalibured]

people heard it called vieta jumping

[zadid xcalibured]

root flipping exercise(for me too)
1.if a,b,$q=(a^2+b^2)/(ab+1)$are integers then q is a perfect square.
2.if a,b,$q=(a^2+ab+c^2)/(ab+1)$are integers then q is a perfect square
3.if a,b,$q=(a^2+b^2)/(ab-1)$are integers then q=5

[Tahmid Hasan]

here's another solution without flipping roots whatsoever but using only descent and inequality.

let $\frac {x^2+y^2+1}{xy}=k$.if $x=y$....(1)
then $x^2 \mid 2x^2+1 \Rightarrow x^2 \mid 1$,hence $x=y=1,k=3$
now let $x>y$,and let $(x,y)$ be a solution of (1) with $y$ minimal.
it is easy to check that $(y,ky-x)$ is also a solution.
now we claim $ky-x<y \Leftrightarrow k<\frac {x+y}{y} \Leftrightarrow \frac {x^2+y^2+1}{xy}< \frac {x+y}{y}$
$\Leftrightarrow b^2+1<ab \Leftrightarrow b(a-b)>0$ which is actually true contradicting the minimality of $y$.
so our assumption $x>y$ was wrong implying $a=b=1,k=3$

[Tahmid Hasan]

root flipping exercise(for me too)
1.if a,b,$q=(a^2+b^2)/(ab+1)$are integers then q is a perfect square.
2.if a,b,$q=(a^2+ab+c^2)/(ab+1)$are integers then q is a perfect square
3.if a,b,$q=(a^2+b^2)/(ab-1)$are integers then q=5

well these problems can also be solved using the same method i've shown without flipping roots.you should try it too .And i learned this technique from "An Introduction to Diophantine Equations" by Titu Andreescu and Dorin Andrica.

[FahimFerdous]

@Tahmid: your proof is called Vieta Jumping if you don't know then.

[zadid xcalibured]

he knows actually.

[Tahmid Hasan]

well Fahim vaiya,as far as i know if the equation $x^2+ax+b=0$ has real roots $p,q$ then by vieta's formula $pq=b,-a=p+q$ and the principle of vieta jumping is based on this and descent.And vieta jumping is also called root flipping accordong to wiki, and the phrase 'root flipping' sounds more 'cruxy' to me

[zadid xcalibured]

root flipping is just one kind of descent.@tahmid just using vieta formula.