I need an hint on how I can prove/disprove a property that I think it's true about a sequence built in a specific way. Let $p \geq 3$ an integer and $n > p$ another integer. I do the following operations, compute $k_0,r_0$ such that
$$ n = k_0p + r_0 $$
then I define $n_1 = 2k_0 + \min(2,r_0)$, I then compute $k_1,r_1$ such that $$ n_1 = k_1p + r_1 $$
and I define $n_2 = 2k_1 + \min(2,r_1)$. For the generic $i$ e compute $k_i,r_i$ such that
$$ n_i = k_ip+r_i $$
and $n_{i+1} = 2k_i + \min(r_i,2)$, I can easily prove that $n_{i+1} < n_i$, but what I want to prove is that for given $n$ and $p$ there's always an index $j$ such $n_j = 2$. Is there any hint you could give me? The only approach I thought was an induction on $p$ but it is not leading me to anything...