Futher to my comment above,I am convinced that the mystery of the sequence of the Primes [prime numbers] can be explained [within elementary algebra] fairly easily.
But first we must take a more careful look at the rules of distribution:
To save writing, let’s put A = addition, S = subtraction, M = multiplication.
Now let us assert that the following four rules are true - simultaneously.
[a] A is distributive over M
[b] S is distibutive over M
[c] M is distributive over A
[d] M is distributive over S
This implies [necessitates] that there exist intrinsic relationships among the algebraic variables - equations [because all the four rules above are valid]. Where rules [a] and [b] fail, this must be regarded as exceptional [there are an infinite number of exceptions].
The above truth of the four rules lead to an algorithm that I have called: The Evolutionary Algorithm. T.E.A is just the kind of algorithm needed to settle this mystery [let’s not call it a mystery, let’s downgrade it to a puzzle - most people like puzzles] of the prime sequence.
However one or two details need to be fixed before this algorithm can be started and implemented. [i] being able to solve the pairs of equations that constitute each step of T.E.A. [ii] there are two [fixed] algebraic variables in the first step of T.E.A. whose values need to be assigned. Once this is gotten past then we should have an algorithm that produces the successor prime, one at a time at each step. Then the manner in which sequence of the primes works may be come clearer, if not absolutely clear. You might have guessed why there are two equations in each step of The Evolutionary Algorithm - its because there are two rules of distribution - rules [a] and [b].
I now want to indicate how T.E.A. is constructed.
The two simplest equations arising from the rules of distribution are:
b.c + a = [b + a].[c + a] rule [a]
b.c - a = [b - a].[c - a] rule [b]
The rules of commutation and association may be considered to be universally valid.
Note the peculiar order in which the terms on the left side of the equations have been written.
The above two equations constitute the first step of T.E.A and are easily solved for the variable: a in terms of b and c. The right sides of the two equations are calling to be multiplied out. There are many actual numbers that can be set for b and c, therby giving a value for: a. However suitable values for b and c in T.E.A. have not yet been found. It could be expeditious to concentrate on uneven integers for b and c [perhaps a few early members from the class: [3, 5, 7, 11, 13, …].
It would be better now to introduce notation.
The first two equations become:
d[1].d[2] + a[1] = [d[1] + a[1]].[d[2] + a[1]]
[ - 1]{2}.[d[1].d[2] - a[1]] = [d[1] - a[1]].
[d[2] - a[1]]
The factor involving minus one: [ - 1] is put there for a good purpose - not to make the maths more difficult.
In the next step of T.E.A. we set d[3] = a[1], which we just have computed in the first step. Then we relabel a[1] as a[2].
The first equation in the second step is:
d[1].d[2].d[3] + a[2] = [d[1] + a[2]].[d[2] + a[2]].[d[3] + a[2]]
The second equation in the second step is:
[ - 1]{3}.[d[1].d[2].d[3] - a[2]] = [d[1] - a[2]].[d[2] - a[2]].[d[3] - a[2]]
We use the second step to compute a[2] in terms of d[1], d[2] and d[3]. Then we set d[4] = a[2] and relabel a[2] as a[3]. I’ll just write out an abbreviated version of the second equation for the third step. The first equation [involving addition] for the third step is easy to write
down.
The second equation in the third step is:
[ -1]{4}.[d[1].d[2].d[3].d[4] - a[3]] = [d[1] - a[3]].[d[2] - a[3]].[d[3] - a[3]].[d[4] - a[3]]
[not really abbreviated at all]
In the step involving a[j] in the second [subtraction] equation, there is a term [ - 1]{j + 1} to multiply the left side and we are tying to solve for a[j] in terms of the fixed [ assigned] values of d[1] and d[2] in step one and the computed values: d[3], d[4], … d[j + 1] from the a[j - 1] values in the previous steps.
The T.E.A. algorithm outlined above uses all of the previous information to find the next piece of information so that the algorithm can be continued one step at a time indefinitely. This is exactly what we need in order to find the sequence of the primes - an algorithm that computes the next prime in the sequence, having used all of the previous primes as input information - once the two hurdles [i] and [ii] have been passed by. Then the sequence can be explained. I have called this basic algoithm: The Evolutionary Algorithm [T.E.A.] because while the multiplicands on the left sides of the equations are fixed in value, once the first two have been assigned [d[1] and d[2] and the others d[3], … d[j + 1] subsequently computed, the actual number of such multiplicands increases indefinitely without limit by one at a time with each sucessive step - hence an evolving [evolutionary] algorithm.
The above insights came to me around February 19th 2011. [If anyone has done the same as above before now, then I would certainly like to know abot this.]
I do hope that my comment is not too long - William P. G. Shaw. [Februay 25th, 2011].