# What is the next number in the sequence ___, ____, 16,256, 65,536?

Jul 31, 2015

The first two numbers in the sequence are ${2}^{{2}^{0}} = 2$ and ${2}^{{2}^{1}} = 4$

The elements of this sequence are ${2}^{{2}^{n}}$ for $n = 0 , 1 , 2 , \ldots$

#### Explanation:

More interesting than ${2}^{{2}^{n}}$ is ${2}^{{2}^{n}} + 1$:

$3 , 5 , 17 , 257 , 65537 , 4294967297 , \ldots$

All such numbers were conjectured to be prime by Pierre de Fermat.

Unfortunately, no Fermat number beyond ${2}^{{2}^{4}} + 1$ is known to be prime.

For example, $4294967297 = 641 \cdot 6700417$

It is (in theory) possible to construct a regular $\left({2}^{{2}^{n}} + 1\right)$-sided polygon using ruler and compasses. No other regular polygon with a prime number of sides is constructible using ruler and compasses.