Famous Problems Where We Only Know the Elementary

Solution 1:

The first four perfect numbers have been known for at least 2300 years. Here are their prime factorizations:

$$\begin{align} 6 & = 2^{\hphantom{2}}\cdot 3 \\ 28 & = 2^2\cdot 7 \\ 496 & = 2^4\cdot 31 \\ 8128 & = 2^6\cdot 127 \end{align} $$

Any knucklehead, looking at these, or perhaps even just looking at the first two, could make the obvious conjecture that they are all of the form $2^{p−1}(2^p−1)$ for prime $2^p−1$.

It's easy to show that all numbers of that type are perfect. (And indeed, Fibonacci's Liber Abaci, published in 1202, makes this observation.) All known perfect numbers are of that type. Euler proved that all even perfect numbers are of that type. But…

Solution 2:

Is $\pi$ a normal number?

All we know is that rational numbers are not normal, and that $\pi$ is irrational.

You can just as well replace $\pi$ by many other constants, and we don't know much about any of them.