Define the sequence
![a_1, a_2, a_3, ...](/media/m/8/e/7/8e7d35432d0963082af493ea5b82e68d.png)
as follows.
![a_1](/media/m/6/1/7/6173ac27c63013385bea9def9ff2b61e.png)
and
![a_2](/media/m/4/0/1/401f4cdfec59fba73ae32fa6769c72cb.png)
are coprime positive integers and
![a_{n + 2} = a_{n + 1}a_n + 1](/media/m/f/e/b/febb27146a3b328d171d32d2242854c9.png)
. Show that for every
![m > 1](/media/m/9/c/a/9ca3daa5ed8f80fbe1cf4a12ee8befd2.png)
there is an
![n > m](/media/m/0/a/5/0a53c4f1b5cd92f4e5dfb3b039240832.png)
such that
![a_m^m](/media/m/f/3/9/f3953bbc676bfb3dfd8e1419bc57b991.png)
divides
![a_n^n](/media/m/6/3/2/632d1d18f33e54c82ae523d1845d2755.png)
. Is it true that
![a_1](/media/m/6/1/7/6173ac27c63013385bea9def9ff2b61e.png)
must divide
![a_n^n](/media/m/6/3/2/632d1d18f33e54c82ae523d1845d2755.png)
for some
![n > 1](/media/m/c/8/9/c8999d29e042cf52e485c7a7b7301b0a.png)
?
%V0
Define the sequence $a_1, a_2, a_3, ...$ as follows. $a_1$ and $a_2$ are coprime positive integers and $a_{n + 2} = a_{n + 1}a_n + 1$. Show that for every $m > 1$ there is an $n > m$ such that $a_m^m$ divides $a_n^n$. Is it true that $a_1$ must divide $a_n^n$ for some $n > 1$?