Solution:
The known primes of this form (Fermat primes) are $3$, $5$, $17$, $257$, and $65537$, and the result is due to Gauss (German). If there are other such primes (unknown), then they are much bigger than $10^{10}$. So for each product of these primes, we can divide $4.3 \cdot 10^{9}$ by that number and take $\log_{2}$ to find the largest power of $2$ to multiply by, then compare the resulting numbers. There are $32$ cases to check, or just observe that $2^{32} = 4,294,967,296$ is so close that there's likely a shortcut. Note that $2^{32} + 1$ is divisible by $641$, and hence not prime. $3 \cdot 5 \cdot 17 \cdot 257 \cdot 65537 = 2^{32} - 1$ is smaller; replacing any of the factors by the closest power of $2$ only decreases the product, and there's not enough room to squeeze in an extra factor of $2$ without replacing all of them, and that gives us $2^{32}$, so indeed that it is the answer.