This year there was a question in the third round of the Iranian math olympiad: if $a,b$ are two natural numbers such that $ab=2^n+1$ and $a,b$ has $l,m$ 1 in their binary representation prove that $n\le (l+1)(m+1)-6$.
the bound is actually sharp for example take $$a=2^\alpha+1,b=2^{2\alpha}-2^\alpha+1,ab=2^{3\alpha}+1$$ But I think it is still not the right bound when $t=min(l,m)$ is big enough. Can you give a better bound for big enough $t$ or give examples with big $t$ that has the same order as the above bound?