Question

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

Answer 1

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,...`

Explanation:

More interesting than `2^(2^n)` is `2^(2^n)+1`:

`3, 5, 17, 257, 65537, 4294967297,...`

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 * 6700417`

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

See https://en.wikipedia.org/wiki/Fermat_number