Because v+20 uses 20 moves to get from v just by additions, but you can solve for v in 15 moves by 15 multiplications. So there is no need to use more than 15 moves of any kind.

I thought it's actually not $$$O(n^{2})$$$ (and yep it should be without log, cause n is already log), but $$$O(n^{\log_3 2})$$$ if you use Karatsuba algorithm. And I heard about some algos, that work like $$$O(n \cdot \log n \cdot \log \log n)$$$ and $$$O(n \log n \cdot 2^{\log *n})$$$.

But it's actually $$$O(\log (a + 1)(b + 1))$$$. ^-^