Can 13, 15 and 17 chameleons become one colour

On an island live $13$ red, $15$ green and $17$ blue chameleons. Whenever two chameleons of different colours meet, both switch to the third colour; nothing else changes colours. Is it possible that at some point every chameleon has the same colour?

No. A meeting changes the three counts by $(-1,-1,+2)$ in some order, which leaves every pairwise difference unchanged modulo $3$. To end with one colour, two counts must both be $0$, a difference of $0 \bmod 3$, but the initial differences are $2$, $2$ and $4\equiv1$, none divisible by $3$.