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1

First of all, they are all numbers.


0

Do you have any reason to believe there is a simple "pattern"? It starts out increasing up to, at least, 3 then increases again to 50. In any case, given any finite sequence of numbers, there exist an infinite number of "formulas" that will give that sequence.


4

The first sequence is twice the second sequence.


0

There is an extremely simple proof of this fact if you simply write the number in binary and realize that $2^n - 1$ is written as $n$ $1$'s. If $n$ is a composite number (i.e. non prime) then it can be grouped into a subset of $1$'s. Here's an example, $2^{10} - 1$: \begin{align} 2^{10} - 1 =&\ 1111111111_2 = 1023 \\=&\ 1111100000_2 + 0000011111_2 ...


1

Your analysis is correct. I don't think it really gives insight into Euler's identity per se, but it does help illustrate some geometric intuition about complex arithmetic, which is a great way to understand Euler's identity. First, $r e^{i \theta} = r (\cos \theta + i \sin \theta)$ represents a point in the complex plane in polar coordinates, with a ...



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