Find all functions
![f: \mathbb{N^{*}}\to \mathbb{N^{*}}](/media/m/b/f/a/bfa36ca77baa808dbff866159d41765c.png)
satisfying
for any two positive integers
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
and
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
.
Remark. The abbreviation
![\mathbb{N^{*}}](/media/m/2/1/e/21ee83eaf965e8810b590d0661acca1a.png)
stands for the set of all positive integers:
![\mathbb{N^{*}}=\left\{1,2,3,...\right\}](/media/m/7/e/1/7e143e4c83663bb1febcba692c60ca85.png)
.
By
![f^{2}\left(m\right)](/media/m/3/a/4/3a48de5a2d1dd58ac4e3bb70e86ab170.png)
, we mean
![\left(f\left(m\right)\right)^{2}](/media/m/a/0/b/a0bd7e49c6d33ccbd341a18131c1feca.png)
(and not
![f\left(f\left(m\right)\right)](/media/m/c/9/4/c9433840c772c6e3f64cd946f8157796.png)
).
%V0
Find all functions $f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
$$\left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}$$
for any two positive integers $m$ and $n$.
Remark. The abbreviation $\mathbb{N^{*}}$ stands for the set of all positive integers:
$\mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $f^{2}\left(m\right)$, we mean $\left(f\left(m\right)\right)^{2}$ (and not $f\left(f\left(m\right)\right)$).