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PRIME NUMBERS and ENTROPY
The first prime number is 2. When we first press the "Start / Restart" button
the second tile is colored yellow (indicating prime), and every 2nd tile after
that is colored blue (indicating not prime). When done with all the multiples
of 2 the first green tile after 2 (ie., 3) must be prime. So it gets colored
yellow, and every third tile after that that isn't already blue (ie., already
a multiple of 2) gets colored nonprime blue. Again, when done with all the
multiples of 3 the first green tile after 3 is 5, so it gets colored yellow,
and every 5th tile thereafter gets colored nonprime blue. And so on and so forth.
Each row has 30 tiles, and 30 = 2x3x5, the product of the first three prime numbers.
Consequently every 2nd column of tiles corresponds to multiples of 2, every 3rd
column to multiples of 3, and every 5th to multiples of 5. So the pattern
generated in finding all the multiples of 2, 3 and 5 is quite orderly - save for
the yellow 2, 3 and 5, all columns are all green or all blue. But the next prime
is 7, and 30 is not divisible by 7, so in coloring every 7th tile blue things
start to look a little unorganized. And it only gets worse for the primes after
7 (11, 13, 17, ...); the entropy of the system increases.
In information theory, given a system consisting of discrete states Xi,
such that the probability of being in the i th state is Pi, the entropy
of the system is defined to be:
Σ Pi log2(1/Pi)
Ok, so here's how I approached the idea of prime numbers and entropy. Before finding
the first prime there is only one state, hence the entropy S = 0. Each prime we
introduce allows us to divide the set of positive integers into more and more sets.
Introduce 2 and there are already an infinite number of sets (states):
(odd numbers), 2(odd numbers), 22(odd numbers), 23(odd numbers), ...
The respective probabilities of a number being in one of these states are
2-1, 2-2, 2-3, 2-4, ...
The entropy of this collection of states and probabilities is S = 2.0.
By the time we get up to the third prime, 5, if we let W = the set of all
positive integers NOT divisible by 2, 3 or 5, then the set of states we have
to deal with take the form: 2i 3j 5k W, i,j,k = 0,..., infinity.
On the assumption I've done the math right, each time we introduce a new prime p
we increase the entropy by an amount
Sp = (log2p)/(p-1) + log2(p/(p-1)).
(You might find this a surprising result, but I found a lot of simplifications and
cancellations.) So the total entropy is the sum of these Sp from p = 2
up to whatever prime we introduced last. As we let this last prime go to infinity,
this sum, the total entropy, also goes to infinity. There are two reasons for this
infinite entropy: there are an infinite number of primes; and their distribution does not die out quickly enough for this sum to be finite.