Fun Fact: 26 is the only number that is perfectly sandwiched between a square and a cube:
\(5^2 = 25, 26, 27 = 3^3\).
Proof Outline: Consider the Diophantine equation
\(x^2 + 2 = y^3\) in the ring \(\mathbb{Z}[\sqrt{-2}]\). This equation just represents the condition that the difference between the square and the cube is 2, so there is one number in between. One can then show that the factors of \(x^2 + 2\) must be perfect cubes. From there finding that the only positive solution is \(x = 5\) and \(y = 3\) is fairly straightforward algebra, though tedious. From that, only 26 has this quirky sandwich property!
History: Fermat, of Fermat’s last theorem fame, claimed to have an elementary proof that the only solutions were +/-5 and 3. Much later, an elementary proof was found by Stan Dolan, but whether Fermat himself had one remains unknown…