Interesting identity arising from fractional factorial design of resolution III

[kalish1]

I am learning about statistical design of experiments, and in the process of mathematically rigorizing the concepts behind fractional factorial designs of resolution III, I derived an interesting equation:

$$k = \sum_{i=1}^{3}{\lceil{\log_2{k}}\rceil \choose i},$$ for which the solutions $k$ are the Mersenne primes.

How can I show this? Is the above equation algebraic? Is it even solvable analytically?

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Thanks!

 

[Aaron Bex]

The equation you have derived is an algebraic equation and it is solvable analytically. One way to show this is to use the generalized binomial theorem. This theorem states that for any natural number $n$ and real numbers $a_1, a_2, \dots, a_n$, the following holds:
$$\sum_{i=0}^n\binom{n}{i}a_i = (a_1 + a_2 + \dots + a_n)^n.$$
Applying this theorem to your equation yields:
$$k = \sum_{i=1}^{3}\binom{\lceil \log_2 k \rceil}{i} = (2^{\lceil \log_2 k \rceil})^3 = 8^{\lceil \log_2 k \rceil},$$
which has the solution $k=M_p$, where $M_p$ is a Mersenne prime.