Crazy Math

...Yeah, and who doesn't have thoughts on the Goldbach Conjecture? It's attracted the gaze of this amateur, and so I'll try to wade into the waters slowly.

There's an interesting sequence in the OEIS -- A144650. I like it because, when you view it as a triangular array, it makes a nice pattern (click on the link, then click on "tabl" in the "Keyword" section). So what about it? These are the indices of the odd composites, taken from the set of all odd numbers. So, if you list them out:

1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25 ...

And pick the values at indices indicated by sequence A144650, you'll get all of the odd composites:

5 = 9
8 = 15
13 = 25
11 = 21
18 = 35

and so on.

This sequence is ordered in a particular way, though, and it's the ordering that's interesting (at least to me). The value at index 5 is 3^2. The value at index 8 is 3*5. The value at index 13 is 5*5. The value at index 11 is 3*7. The value at index 18 is 5*7. Shown visually, you can see that we're reading the columns out of a product table:

	3	5	7	9
3	9	15	21	27
5		25	35	45
7			49	63
9				81

O.k., you might ask, so what's the big deal? The fun part of this is it suggests to us a way to convert the Goldbach Conjecture in to a Goldbach Property -- in other words, we can devise a testable statement that determines whether or not any set of positive integers S can be used to construct a sum table that contains all of the even positive integers.

Let's go back to our set of odd numbers. I'm going to give the set a name, A:

A = {x: x>0, x == 1 mod 2}

Which is a fancy way of saying the set A contains all integers (greater than zero) such that x == 1 mod 2, or x is an odd positive integer.

We can construct a number of functions which generate the individual terms of A, but I happen to like this one:

ax - a + 1 = n, a = 2, n is included in A

Or, simplified (after replacing "a" with "2"): 2x - 1 = n


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Now for our first trick. Let's define a set B to be all the even integers:

B = {z: z>0, z == 0 mod 2}

And a function to generate its members:

az - a + 2 = p, a = 2, p is included in B

It is pretty easy to show that the sum of two values from set A will be included in set B:

ax - a + 1 + ay - a + 1 = p, p is included in B

Simplify:

ax + ay - 2a + 2 = p.

Let a = 2:

2x + 2y - 4 + 2 = p

2x + 2y - 2 = p

And it follows from inspection that this will always produce a value p such that p == 0 mod 2.

So, we can restate the equation for generating set B from values of set A as:

ax - a + 1 + ay - a + 1 = az - a + 2

Let's simplify; first gather all the variables and constants:

ax + ay - 2a + 2 = az - a + 2

Subtract 2 from both sides:

ax + ay - 2a = az - a

We can now divide through by a, removing it entirely from the equation:

x + y - 2 = z - 1

x + y - 1 = z

Interesting... What does this tell us? Remember we defined the equation that generated all values of A as:

ax - a + 1 = n, a = 2

But now the value of a doesn't matter... That means we could use any value we wanted for a and end up with the same result. For instance, let's set a = 5. Plugging into the equation above we get:

5x - 4 = n

This, in turn, will generate a new set of integers that I'll call C:

C = {1, 6, 11, 16, 21, 26, ...}

Remember, though, that we also defined a set B, which in the case of a = 2 was the even integers. Well, we can now generate a new set, which I'll call D, using the same equation for generating B but setting a = 5:

5x - 3 = r

D = {2, 7, 12, 17, 22, 27, 32, 37, ...}


Here's the fun part -- sets C and D are related to each other in the same ways as A and B. First:

let x, y be included in A, then:

x + y = z, z included in B

x * y = w, w included in A

What this means is that if I add together two elements of A, then I'll get an element of B. If I multiply two elements of A, then I'll get another element of A.

The additive relationship is the basis for the Goldbach Conjecture, and the multiplicative rule is the basis for determining primality.

Primality:

Since x * y, where x and y are included in A, is also included in A, it allows us to define the concept of "relatively prime". Meaning, in other words, that if a value of A cannot be stated in the form x * y, with x and y both included in A, then that number is relatively prime with respect to A. Remember, A is the set of all odd numbers. So, it follows that all odd numbers that are prime have no divisors in set A, and we can check this exhaustively by dividing any member n of A by all the values of A less than the square root of n.

Goldbach:

Now that we have a way to define primes in the set A, we can restate the Goldbach Conjecture as a statement about set relationships. Let:

A = {n: n == 1 mod 2}
B = (q: q == 0 mod 2}

Take x,y such that x and y are included in A and are relatively prime to A. Then all the members q of set B can be expressed as the sum x + y = q.

Now that we've defined what "relatively prime means, we now have a way to define the Goldbach Property in terms of relative primality:

let X be the set expressed by the equation ax - a + 1 = p
let Y be the set expressed by the equation ay - a + 2 = q

Then, the Goldbach Property is:

The property that, if a_i and b_j are included in X and are relatively prime to all members of X, then any member q of Y can be expressed as a_i + b_j = q, for not necessarily distinct values of i and j.

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What does this mean in practice? Well, I'm not entirely sure. For one thing, by picking different values of a we can generate an infinite number of sequences that look and behave like A144650:

a = 2, sequence = 5, 8, 13, 11, 18, 25 ...
a = 3, sequence = 6, 10, 17, 14, 23, 32 ...
a = 4, sequence = 7, 12, 21, 17, 28, 39 ...

And so on (and in both directions, i.e. a = 1, 0, -1, -2 -- more on that later). What does that mean, then? Note that these are the indices of all the "composites" within each sequence. By exclusion, we can be sure then that any value at an index not in these sequences is "prime". As the sequences get bigger, so do the number of primes in a sequence.

Which is all a really long way of getting to the point -- at what value of a can we prove the Goldbach Property for the sequence generated by a? The distribution of relative primes is changing, as we go from sequence to sequence, in a very regular sort of way, thus I presume that all the machinery arrayed against the classic Goldbach Conjecture will also fail to prove whether or not any sequence generated by any value of a fulfills the Goldbach Property as I've defined it here. I don't know if that's entirely true; most of the efforts at proving Goldbach have involved demonstrating that some statistical function produces values within a certain range of bounds, which then translates into statements like "every even integer is the sum of, at most, 20 primes".

So, perhaps, we have a new way to attack the problem. Go waaay back near the top and notice that the value for a ultimately doesn't matter:

x + y - 1 = z

This is ultimate a statement about the indices -- if you take the indices x and y from A, add them, then subtract 1, you'll get the value at index z from set B. Let's test this:

x = 2, y = 3:

2 + 3 - 1 = 4

Which means:

3 + 5 = 8

and, indeed, by inspection we can see that from set A:

{1, 3, 5, 7, 9 ...} the values at indices 2 and 3, when added together, equal the value at index 4 in set B:
{2, 4, 6, 8}

Applying this to the Goldbach Conjecture, then, we're really concerned about whether or not the set of prime indices "codes" for all the integers. In other words, let:

P = {2, 3, 4, 6, 7, 9 ...}
P1 = P_i - 1 = {1, 2, 3, 5, 6, 8...}

Let x_i be included in P and y_j be included in P1. Then we want to show that:

x_i + y_j = z, for all z included in Z, where Z = {n: n > 2, n is an integer}

As we range over all possible values for i and j.

Anyways, some food for thought...