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January 26, 2024

Concept sharing and Goldbach’s conjecture

Filed under: goldbach's conjecture,Mathematics — Rob burchett @ 4:26 pm

Abstract: Here I explain and use concept sharing. I take a look at Goldbach’s conjecture.

An interesting notion can come about from something observed in nature, the overlapping of two shadows. Consider a cup of tea placed on a table and lit with two lights from above, one from the left and another from the right, as illustrated below:

Three areas of shadows are formed. The one in the center is the overlapping of two shadows.

The concept sharing of a number:

Numbers are exact concepts. In the above case, we can think of them as the number of overlapping shadows at the center. They have exact boundaries and some way of showing we have two overlapping shadows there or three there, etc.

Then borrowing from the notion of overlapping shadows we should be able to hide numbers together and they would be “two hidden as one” as well. (concept sharing) if the mathematical objects represented by the numbers had the same boundaries, like the shadows at the center.

Other than the further darkness of the overlapping shadows, we cannot see or imagine that there are two separate shadows there. Similarly with two numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.

Then to this end let us create another number dimension, a dimension of number of numbers. Let the usual case be that the number of numbers is only 1. But now let us expand into the next dimension and allow the number of numbers to be 2.

So for example with the number 1, let us take away the original number 1 (since we have another underlying dimension, we can do this) and replace it with two new numbers 1(1) and1(2). These are together like the two shadows but do not form one number. The number then has a one-dimensionality to it as it is at least possible to separate it linearly.

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects hidden as one. Or you can think of the two number 1’s as fitting into each other.

Since numbers are exact we can consider the objects to be like points. Since something with no extent placed onto something with no extent would still have no extent but there could be two objects here. Also we could think of a combined single object of no extent( at the start) created form two other objects of no extent.

When we are numbering two mathematical objects which fit into each other, the objects are somehow different from each other and we give the two hidden objects two new numbers 1(1) and 1(2).

In the case of mathematical objects there is no external way of telling how many objects there are, previously it was assumed it was only one. We can state how many we wish at the onset thus fixing a certain mathematical system. Then we need the concept sharing of a number to indicate how many objects we wish to be there.

Goldbach’s conjecture:

We can think about new types of numbers (1(1)(1(2)), (2(1)(2(2)), (3(1)(3(2)),..These are higher numbers than the naturals, 2 being greater than 1. The notation is meant to show that the numbers are together in the number space, fitting into one another or hidden as one. I show a revelation of one number “peeking out” from concept sharing with another number. Shown more clearly here: (a(a). For clarity and brevity I drop the (1),(2) designation from now on.

There are other numbers such as (2(3), (3(4), (3(5),… We can think of this as partial sharing. For (2(3) think of three dots colored red. We can have two blue dots overlapping with two red dots forming two purple dots. I say that to give the idea, numbers are not colored dots. Numbers are the exact concept of amount or position only. So that (2(3) can be thought of as two sharing with three and (3(2) can be thought of as three sharing with two. In the first case 1 is left over and in the second case 1 is an extra number. These are the same concept though so (2(3)=(3(2).

Consider the even shared numbers (2(2), (4(4), (6(6),…

We can break them down into a sum of two prime shared pairs such as (2(2)=(1(1)+(1(1) and also (4(4)=(1(3)+(3(1) if we define the other types of shared numbers with dissimilar basic numbers to fill in the rest of the possibilities. They can form products too.

Some numbers have more than one decomposition. Such as (16(16)=(13(3)+(3(13) and (16(16)=(11(5)+(5(11).

Now think of the even shared numbers as being created from the primitives. I designate them (2(2), (4(4) … but these numbers are not created from the usual numbers on the number lines, as these are lower numbers, these higher numbers; (2(2), (4(4)…. must come from somewhere else. These even shared numbers must go on indefinitely, as to lead to the numbers 2,4,6,…at the lower level . Then the natural numbers are all at a lower level from these numbers.

For multiplication, in the usual number system we have prime numbers. We can look for the higher level of prime numbers. Suppose I build a higher level of primes from the usual prime numbers by concept sharing two prime numbers. For example (7(3)=(3(7).

But (10(10) = (7(3)+(3(7) =(2(2)(7(3). (7(3) can’t be equal to (7(1)(1(3) as I can change this to (7(1)(3(1)=(21(1). The other part, (3(7) can be changed to (1(21). I can add to obtain (10(10)=(22(22), a contradiction as 10 does not equal 22. This is outside the sharing domain. So (3(1) and (1(7) can’t be factors. So I can’t factorize (a,b) further where a and b are prime.

Then starting at (4(4), we can ask, how is (4(4) created? (4(4)=(2(2)(2(2). (4(4) is an even shared number so there is a division by (2(2) possible. This can be the definition of an even shared number. When I divide both sides by (2(2) I have a shared number on both sides, these are equal but not necessarily identical. For (4(4)=(3(1)+(1(3)=(3(1)+(3(1) =(2(2)(3(1) also. In this case we can think of the shared number (2(2) separating in the number space to form the number (3(1)=(1(3).

Then also (6(6)=(2(2)(3(3). Also (6(6) =(5(1)+(1(5) =(5(1)+(5(1) =(2(2)(5(1) also.

Also (8(8) is not equal to (2(2)(4(4) as (2(2)(4(4)=(2(2)(4(2)(1(2)=(2(2)(2(4)(1(2)=(4(16). This is outside what we could consider the sharing domain as 8+8=16 but 4+16=20. We find (8(8)=(5(3)+(3(5)=(2(2)*(3(5)=(2(2)*(5(3)=(10(6)

And so on. Each even shared number must be a multiple of (2(2) and another prime shared number. We consider 1 to be a prime number.

Imagine an extended number line. For example, since (5(5) can separate, lets look for a way (10(10) can be created. (10(10)=(a(b) +(b(a)=(2(2)(a(b). I am taking apart (5(5) so as to create (10(10). Two (5(5)’s are separated and then recombined. ex. (7(3)+(3(7). (10(10) can be created from (a(b) +(b(a) but this is also (a(b)+(a(b) =(2(2)(a(b) so this must work too. As shown in the case above with (8(8), a,b must both be prime. There must be one such factorization.

Every even shared number can be divided by (2(2). The other factor must be a prime shared number as we need this to work in the shared number system. For example (12(12)=(2(2)(6(6). (6(6) must separate into two prime numbers otherwise (12(12) can’t appear (we assumed it was there). It must exist since I need to have 12. I can now take (12(12) out and replace it by 12. So we must have a decomposition with (2(2)(a(b) with a, b both being ordinary prime numbers.

This demonstrates the equality of every primitive decomposition as well (which the lower level was hinting at) since for any even shared number there may be more than one decomposition. We go back to form (16(16) and then can separate once again to find another decomposition. Here we find two, (2(2)(13(3) and (2(2)(11(5)

Then if we look at one half of these binary decompositions, that is, for example, look at 16(16)=(13(3)+(3(13) and see 16 =13+3 we can see Goldbach’s conjecture is true. It was just a part of a deeper understanding of numbers.