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An alien civilization has a strange way of notating their numbers. Here are the first 30 numbers in their system:
The notation is unique up to slight differences in handwriting. Therefore you may ignore tiny details/mistakes such as the ink splotch on the right of the 14. Now tell me what this number is, and why:
We can read each number as follows:
The number 0 is represented by nothing at all. A single dash represents the number 1.
If there's no connector at the top, it's a prime power:
The prime index is found at the tip of the arc - read the number recursively, then move this many primes forwards. In the examples above, we see that the number 29 is encoded as p_9 because it's the tenth prime, number 2 being p_0.
The power is found at the bottom of the arc - also to be read recursively. In the examples above, we see that 27 is encoded as 3^3.
Composite numbers are encoded as the product of their primes factors, starting with the smallest prime. In this case, the prime indexes for subsequent primes start at the next prime after the one to the left of it. We can see it in number 15, which is encoded as p_1 * q_0.
As for the number the second image, it's
p_{3^2 * 11} = p_99 = ...
require 'prime'; Prime.take(100).last
... = 541
I'm going to guess that the answer is
541
Explanation:
Placing $A$ on top of $B$ gives $A^B$.
Placing $N$ below and to the right of the symbol for "2" gives $P(N+1)$, where $P(n)$ is the $n$th prime.
Placing $A$ and $B$ next to each other with a bar above gives the result of a somewhat complicated function. If $A$ is represented by $p_1^{n_1}p_2^{n_2}\ldots p_k^{n_k}$ and $B$ is $q^m$, where $p_1 < p_2 < \ldots < p_k < q$, then the result is $A \cdot P(P^{-1}(q) + P^{-1}(p_k))^m$. That is, $B$ is a prime power, and its base is increased by the index of the greatest prime divisor of $A$ before multiplying by $A$.
The last step in the answer in the image below is wrong -- it should be $P(99+1) = P(100)$.
And here are my notes:
Partial answer to begin with: it appears that
if a number can be written as $x^y$, then the character for $x$ is written on top of the character for $y$. See $8 = 2^3$, $9 = 3^2$, $16 = 4^2 = 2^4$, $25 = 5^2$, and $27 = 3^3$.
The sixty-third prime, which is 307 according to the first website I saw. Prime numbers greater than two are expressed as a "branch" from the middle of the curve signifying "2". )-
The line from the top of the curve means multiplication by the prime number in the series proceeding the prime it appears to be. ) 9*7 means 63rd in the series of primes.
