Fusible numbers are a set of numbers defined by Jeff Erickson as the periods of time that can be measured by burning ropes that each take 1 unit of time to finish burning. You can only measure the time between the first burn and the completion of the entire burn. You can wait for a given rope to finish burning before starting to burn another one, but you cannot assume the ropes burn at a uniform rate.
It is known that the set of fusible numbers are dyadic rationals, and that they become very dense on the number line when measuring longer periods of time. The smallest fusible number possible is $\frac 1 2$, the smallest fusible number above 1 is $1 \frac 1 8$, and the smallest fusible number above 2 is $2 \frac 1 {1024}$. However, the smallest fusible number above 3 is not known, only that it is between 3 and $3 + 2^{-2 \uparrow^9 16}$.
There exists a recursive formula to compute the gap between a fusible number and the next larger one, it is mentioned in https://arxiv.org/abs/1202.5614, which also mentions that a more well-known formula from Erickson himself is actually incorrect. But this is not practical for finding the gap between 3 and the next fusible number, since computers cannot handle numbers as large as $2 \uparrow^9 16$.
I am wondering whether it is practical to find the smallest fusible number above the number $2\frac 1 2$ instead. This seems like it could be a feasible value to compute, as I naively implemented the algorithm above in Python and could determine that the smallest fusible number above $2 \frac {13} {32} (\approx 2.40625)$ is $2 \frac {13} {32} + 2^{-31}$.
Though, the density increases rapidly and non-monotonically, so I would not be surprised if $2 \frac 1 2$ is already out of reach. After all, my script had more trouble computing the smallest fusible number above $2 \frac {255} {1024} (\approx 2.24902)$, than it did computing the smallest fusible number above $2 \frac {13} {32}$.
If $2 \frac 1 2$ is indeed too much to handle, I wonder what bound on its gap size can be proven, ideally a bound that is so small that it clearly demonstrates the impracticality of computing the value directly.