Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^2+b^2$ ; then prove that $\frac {a^2+b^2}{ab+1}$ is a perfect square (this problem came in $\Bbb {IMO}$ $1988$). How to prove it without using geometry or induction ?
Asked
Active
Viewed 287 times
1
-
4Search the MSE site for IMO 1988. You will get a number of hits, several highly relevant. – André Nicolas Apr 25 '13 at 05:44
-
3Would any of the dozen solutions posted and linked to e.g. here answer your question? – Jyrki Lahtonen Apr 25 '13 at 05:45
-
@JyrkiLahtonen: Uh,thank you. – Souvik Dey Apr 25 '13 at 05:48
-
1NP. This is a fun problem! It is just kinda famous :-) – Jyrki Lahtonen Apr 25 '13 at 05:51
-
please check this (second proof) http://math.stackexchange.com/questions/28438/alternative-proof-that-a2b2-ab1-is-a-square-when-its-an-integer – Lotusquantum Apr 25 '13 at 05:55
-
@JyrkiLahtonen: Yeah I know , I just now proved it by extremal principle. – Souvik Dey Apr 25 '13 at 05:56
-
New and better solution without using vieta jumping method here http://math.stackexchange.com/questions/28438/alternative-proof-that-a2b2-ab1-is-a-square-when-its-an-integer/646382#646382 – MathGod Jan 23 '14 at 06:56