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Hide and seek is a fun game. But suppose instead of children, numbers would like to hide.

Suppose you are the number 1. Where could you hide? Numbers are so obvious. They go 1,2,3,4…etc. How could 1 hide?

But suppose I have two number 1’s. I could hide both together, by putting them together so that the two of them could not be seen, they just look like one number 1.

LIke so 1…..>1(x2)<……1.

We have two number lines, perpendicular to each other, as shown. There is a way of moving the two 1’s so that they can be placed together and not be one 1, this is shown later, below.

We would need another level to the number 2, as these two items here are in the same place. These two items just “look like” one item. We cannot imagine two items there, but our logic tells us this could be so. Numbers are in a sense zero-dimensional.

Since we can have this, there is no way of knowing how many are hidden here ahead of time. Simply state ahead of time how many we want.

We need to take away the number 2 and replace it with a label like (1(1)(1(2)), showing that there are two items there, yet the two items do not combine to form one, 1. The two new label 1’a share the concept of the label, 1.

The space is a space where a point can be multiple.

Since it is logical that two items of no extent can exist together and be two items, it must be that the usual idea of a point is replaced with this new idea, so as to give it existence. See the teacup shadow diagram below. Also places can be moved off in this new dimension (can be separated) Hence they are different from each other and different from the usual 1, because they can be separated.

A point is the notion of that which has no extent or location only, labelled, p.

But since the idea of no extent is to have no dimension.

Two ‘items’ of no extent could exist together and still have no extent The two will be hidden as one.

It would be two of the same location.

Since points have location only, this could not be two points, but would have to be two of something else, call them ‘parts’.

Represent them by the letter e.

e has no extent but can also be multiple or can concept share. The concept of no extent is shared by the e’s. We could notate two of them together as e(1)(1)+e(1)(2).

When there is only the idea of no extent, without this natural extension two points can only combine to form 1 point. But with this extension 2 points can exist together as 2 points

The concept of an item of no extent is basic to geometry, but what if there were more than I item of no extent? As you shall see, this maybe non-intuitive but it’s not so far-fetched.

What if two e’s(items which are like points but not exactly points) could be placed together so that they do not form 1 e. ( otherwise they would be the same as points). Additionally points, which we could call p, do not combine with e’s (otherwise e’s would be p’s).

So I say two e’s together, but these two e’s are hidden as one. There might be 3,4,5, etc. We have to say at the outset how many we would like and eliminate 1 and all the other possibilities as well.

With just the simple plane two moving points can only combine to form one point, but now that we have extended geometry, two e’s moving in the new plane can form this new non-combination (e(1)*e(2)).

This also hints at another concept sharing, that of a concept sharing of a number. The usual number of 2 does not really apply since the two e’s are together so we take out the number 2 as the only form of 2, like we did with the simple point and seek a lower level of number of original numbers. Now let it be two instead of one so that we might label the two e’s 1(1) and 1(2). The idea being two 1’s in a tie, not forming one, 1.

How are 1(1) and 1(2) different? The two labels are different from each other and from the label 1 (usual label)

Since I can move them I can place them in a new unit of one, after I replace the 1 that’s there. Then the usual number line can be thought of as the diagonal line, with the associated 1 being non-movable (the idea of a fixed unit of 1). We are able to remove the usual 1 that’s there and replace with 2, new movable 1’s.

Why do we want to have two number 1’s? We can extend numbers into a number plane and see an underlying level of numbers. The symmetry of these numbers can clear up old problems, since we are looking at an underlying level to numbers.

We only have to take the usual 1 that’s there out of the way and two number 1’s can be where that 1 was.

To do this we need another level of numbers like a one story house has a basement. Then I could remove the number 1, like removing the first story, and still have the basement. Then build a two story house on top of the basement that’s left over.

The two number 1’s sharing is an example of Concept Sharing.

Mathematical Concept removal and sharing

Introduction:

The notion of a point, that which has no parts or no extent, is basic in math. The ancient Greeks thought about points, but what if they were not entirely correct?

They asked, what if two points were placed next to each other? They thought that this would be one point and stopped there.-Aristotle : Physics- “neither can two points be contiguous with one another”

But what if this could be regarded as two points? Since something of no extent placed next to something of no extent would have no extent- but there could be two items of no extent there! The two points could be hidden as one!

Yet, as Aristotle thought the two indivisables of no parts would overlap for the two items can be considered together, side by side or to be on top of one another. This is possible with items of no extent as we can consider them either to be on top of each other or side by side. If two dots are placed side by side, they can be considered on top of one another if we look at them from the side. Also if two dots are placed on top of one another we can look at them as being side by side from the top.

Since we want to keep this concept of no extent, or no parts it must also be possible to have two together and still have two, as long as the two parts are different entities so that they do not combine into one. That is, there must be another possible entity with no extent!

One can regard the overlapping shadow diagram below:

Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:

Now as seen in the overlap, two different points of the shadows can take up the space of one point, we can regard these as e( e is another possible entity of no extent)  and p and not points, p and p. If I take away one light, one shadow still remains. 

This concept cannot be visualized as one would be trying to visualize two different items in the same place, but it is still logical, nonetheless. Even the concept of no extent cannot be fully visualized as well. Only usually we represent a point with a dot, but this is not really a visual of an object with no extent.

Also, consider that we have to take a point out in order to have this e*p present. That is e, the new entity sharing position with the point, p. That means we need another level to place, this must exist since we have to have e*p somehow.

Then this is how e can be made different from p, as in an extent of e and p which can come about, we can move e through this e extent, it then has a new location of original location, making it different from p. That is, e is the next level of location.

Yet we want this concept of point removal and sharing (the point is removed and the two points are sharing the notion or concept of an object of no extent) to join up with the knowledge that is already in existence.

In doing this we have to add to and change the mathematics which already exists. This turns out not to destroy but heal mathematics, as this can be seen as something which is basic, but is missing. The math from this leads to a way to solve many old problems which are easy to state but previously hard to solve, such as:

The knot problem:https://calctutor.ca/category/mathematics/knots-mathematics/the-knottedness-and-chirality-of-the-trefoil/

The twin prime conjecture:https://calctutor.ca/category/mathematics/the-twin-prime-conjecture/

The Goldbach conjecture:https://calctutor.ca/category/mathematics/goldbachs-conjecture/

Fermat’s last theorem;https://calctutor.ca/category/mathematics/a-clearer-and-simpler-demonstration-of-fermats-last-theorem-wiles-theorem/

The Poincare conjecture:https://calctutor.ca/category/mathematics/the-poincare-conjecture/

The Riemann Hypothesis:https://calctutor.ca/category/mathematics/the-riemann-hypothesis/

The Collatz conjecture:https://calctutor.ca/category/mathematics/the-collatz-conjecture/

In fact this idea can complete mathematics:

The completion of mathematics:https://calctutor.ca/category/mathematics/the-completion-of-mathematics/

Definition and description of e:

When two points come together or start together, it seems as if there could be two there as 0+0=0.(two items of no extent could still combine to no extent). Only that points have location only, and having them together means we have one location so we have to conclude that there is one point there.

What if an object of no extent could be expressed as two different items of no extent, starting together. Since an item of no extent together with another different item of no extent would still have no extent, yet there could still be two items here and not one. The other item of no extent would have to have something more than location (it would have to be different from a point).

Yet since it is logical that two items of no extent could be together and still be two items, this must be mathematically possible, somehow. It would work if the two items were not both points, in the sense that they were not the same type of entity. For points are defined to have location only and here we are at the same location.

Call this new indivisible item “e”. Then this together with a point can be called e*p. We can separate e and p (not divide as this is a separation not a division) if we move the original e along an extent made up of e’s. This extent of e’s can be together with an extent of p’s. The e extent is different from the p extent in that it is a series of positions of the original e position. That is it is the next level of position.

Then we have to have an object of no extent which is somehow different from the usual conception. I am thinking of putting two items of zero size together. One can think of putting a dot on top or beside a dot which is there already. One cannot visualize two items of no extent being together, but one can also not visualize an item of no extent as well.

Yet they are an e( new entity) and a point (p) at the same place, if we have a preexisting space made up of ordinary points (p’s). 

In terms of symbols, one may write pop=p as the idea that for an ordinary point two points starting together or being brought in coincidence form again an ordinary point. O is the idea of coincidence.

Then e would be something different. E then cannot be in coincidence with p, if we redefine coincidence as the starting together or coming together of two or more points. so we invent a way for the entities to be together and call it sharing, E*p is e sharing with p . Sharing is a way for two items to come together or be defined together and not combine. This should be possible as seen in the analogy of the overlapping shadows. This must exist somehow in mathematics, since we must have the notion of something with no parts or no extent and we can put two together and still have two. So we have p and e as different entities which are not identical. 

Additionally we must remove p for p*e( p and e together) to exist otherwise there is confusion. If we start with a space with P’s everywhere we have to remove a p to put in an e*p. So there must be another level to space. A container of points and e’s. This would then be a new place of original places, another level to place. Then we can see that e can be different from p, as in an extended space of e’s, I could move an e to another new place of the original place of e, along the e continuum. The e continuum can coexist with the p continuum.

So e is another type of point. It is able to be with another entity which has no extent and not be equal to that entity. P doesn’t share with p, as this is already seen as coincidence. Since points have location only this is the same location and so the same point.

Looking at e, e(1)oe(1)=e(1). But also possible is e(1)(e(2)), if we have an extent of e outwards from p(1). e(2) is some e on the e extent outwards from p(1). This is what makes e different from p, with p we only have pop=p. That is an e can act as a type of container for other e’s, as two different e’s do not combine similar to a p and an e. That means we can have a shifting movement of one e along another continuum of e’s. Then e’s can have a new location of the original location of an e which makes them different from p’s, which have location only.

e(1)*p(1) is e and p sharing a location. We can separate a copy of e(1) away, since 0-0=0. Think of a jigsaw puzzle of a landscape being taken apart. e(1)*p(1) =e(1)oe(1)*p(1). Let e(0) be a bridge into the other dimension. Then e(0)(e(1))oe(1)*p(1)=(e(0)*e(1))o*p(1) and p(1) falls away leaving e(0)(e(1)). Then we have e(2)(e(1)) with e(1) moving to an e(2). There might also be an extent of p, along with the e, In which case we can have e(1)*e(2)*p(2).

The e can be multiple, also e can move along another extent of e’s. This is how e is different from p.

To be complete…=p(1)op(1)op(1)op(1) =p(1)op(1)op(1)=p(1)op(1)=p(1), and also….=e(1)oe(1)oe(1)oe(1)=e(1)oe(1)oe(1)=e(1)oe(1)=e(1) and also.. e(1)*e(2)*e(3)*e(4) is not equal to e(1)*e(2)*e(3) is not equal to e(1)*e(2) is not equal to a single e .Also e(0)(e(1)*e(2)*…)). That is, e or multiple e’s can travel along an extent of e’s.

These would not be points, then, as points have location only. Two moving points could come together to form one point. In Physics or mathematics if we have a plane of points(fixed) there can be a point moving in this plane which takes up the locations of these planar points as it moves, and then we have pop=p as it moves.

But also e’s can move as well and we could have two e’s come together to form e*e when the initial location is moved to the extent of e’s.

A point is the notion of that which has no extent or location only, labelled, p.

But since the idea of no extent is to have no dimension.

Two ‘items’ of no extent could exist together and still have no extent. The sum of any number of these new items of no extent is also no extent. The two will be hidden as one. They are hidden as one, at the ordinary point in the pre-existing space. 

It would be two of these new locations at a regular location. By regular location, we mean an ordinary point of the pre-existing space. 

Since points have location only, and not this multiplicity, this could not be two points, but would have to be two of something else, call them ‘parts’.

Represent them by the letter e.

e has no extent but can also be multiple. E’s have the position to them given by the pre-existing points of the pre-existing space.

Two little nothings still add up to nothing, yet there can be two little nothings there. The two items would just be hidden as one. Let’s hide!-they say. Who can see us? The two items would not be points, for points have a singularity to them. That is, when I combine two moving points, in the usual geometry, it leads to one point, but this doesn’t have to be so with another possible entity. 

These two items could be multiple. This is how we can hide! In order to do this we would have to take out the usual concept of a point as being the only entity with no extent and replace it with these new conceptions as other entities which can have no extent. There are these little entities that don’t combine to form a point. This is because a point is already there as the only idea of no extent. We need some room!

If you accept the notion of an entity with no extent you have to accept this new possibility as well. If you want a point, you have to have us too! The new entities say! The notion of no extent leads naturally to concept sharing! The concept of no extent is an example of a concept.

To remove the concept of a point as the only item of no extent, we need to have a concept space, consisting of a concept of multiplicity. This must exist because I must have this further concept of a doubled item of no extent somewhere and I need to take out the concept of a single item as the only item of no extent to have it appear somewhere. Also we must take out any other possibility of any number of items, except 2.

Extension to other concepts:

Furthermore, all math concepts are point-like in that they are exact ideas which have no existence in physical reality. They are mental constructions which are not available to the senses. They are in thought only.

We can replace e and p, as two items sharing the concept of no extent with two items sharing the concept of a number as well. Or the concept of a set, or group, etc.

Math concepts are exact so that they don’t change from person to person or over time or space. A number, point, function or group, etc. is unchanging, eternal, unalterable.

So also they all can be multiple as the basic ideas of math are all based on geometry and numbers. Functions map inputs to outputs based on formulas which are algebraic expressions of variables(numbers) to points on a graph. Groups are collections of the rotations or flips of geometric objects. Elements of groups are exact in the end numerical or geometric. So numbers, sets, groups, functions, etc. can all have concept spaces!

The idea is to extend the concept using the idea of the concept and the extender “of”. So for example; location of location, number of numbers.

The idea is to go backwards into the idea.

Look at the example of location. For a new location of an initial location we need an extended location, that is there must be an extended location, somewhere to put one of the entities, the extended entity.

Then the extended entity has the location of an initial location, which makes it different from a point, having only location.

The concept sharing of a number:

We can start with a line of p’s with two lines of e’s sharing locations. All in a plane of e’s as shown below:

The numbers 1, 1(1) and 1(2) can be associated as shown, an e is removed and replaced with an e*e and we can associate the new numbers 1(1) and 1(2), concept sharing with the number 1. The number of numbers is 2 and not 1. We can then unfold the two lines of e’s to make an axis where we can have the numbers 1(1) and 1(2). The two numbers 1(1) and 1(2) are sharing the concept of 1, with 1 on the number line. We have the usual number line on the midline of the new plane.

This, then is how the numbers move, they are associated with moving e’s. See below:

The locations move off into the new plane. The numbers come along with the moving e’s. We can have one e at 1, two e’s at 2, three e’s at 3, etc.

Now since two number 1’ s are playing hide and seek, how do we tell that it is not the case that we have the usual case of 1 number 1?

We can show that there are really 2, 1’s there by calling it (1(1) like so:

(1)(x2) = (1<….(1) =(1(1). But what are the numbers representing? The number of dots. As shown below:

The left 1 is really together with the right 1, yet I need to work with both of these so both of these are shown.

So not only would 1 like to play this game but the other numbers would like to join in. 2,3,4,5,6,…etc would also like to hide.

So we can have (2(2), (3(3), (4(4)…

Then also we can have the other games that numbers like to play like addition (+) and multiplication (x).

So (1(1)+(1(1)=(2(2) and (2(2)x(3(3)=(6(6) for examples.

These numbers can all be on a line. The hidden number line:

But now 1 says I would like to hide with 2. Since part of 2 is 1 I could hide with part of 2. Then this could be labeled (1(2) and it would be equal to (2(1). We just have a smaller number hiding with a larger number. 1 and 2 are in the exact same place. The number of numbers is two.

But where are these hidden numbers on the hidden number line?

The answer is that these numbers are on a number plane! A plane of numbers that looks like the diagram below:

On the midline we have the usual hidden numbers.

Then we can ask, how can these hidden numbers play the usual games of numbers? (addition and multiplication)

Let’s investigate!

(1(2) + (1(2) =(2(4)

But also,

(1(2) + (2(1)=(3(3). So it appears that in order for (1(2) to play (2(4)=(3(3). But there’s nothing to say this isn’t okay, since there are no rules for these partially hidden numbers-yet!

Goldbach’s Conjecture:

On June 7th, 1742 Christian Goldbach wrote a letter to Leonhard Euler In which Goldbach guessed or “Conjectured” that every even number is the sum of two prime numbers. So for example: 16=3+13.

Let’s look at the guess in terms of the hidden numbers!

We can think about new types of numbers (1(1)(1(2)), (2(1)(2(2)), (3(1)(3(2)),..These are lower numbers than the naturals, the idea of concept sharing being deeper than the usual idea of concepts. 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: Like so (a)x2….(a<……(a) leading to (a(a). For brevity I drop the (1),(2) extra 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 (4(4), (6(6),…

We can break them down into a sum of two prime shared pairs such as (4(4)=(2(2)+(2(2) if we define the other types of shared numbers with dissimilar higher numbers to fill in the rest of the possibilities of a number plane. 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 by multiplication, similar to the way composite numbers are created from primes.

I designate them (4(4) … but these numbers are not created from the usual numbers on the number lines, as these are lower numbers, these lower numbers; (4(4)…. must come from somewhere else. That is I do not choose 4 and 4 and put them together. I create (4(4) from more primitive numbers and then to get to 4, I have to replace (4(4). These even shared numbers must go on indefinitely, as to lead to the numbers 1,2,3,…at the higher level . Then the natural numbers are all at a higher level from these numbers.

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

Prime numbers are numbers which have 2 factors, 1 and itself. 1 is not a prime number as it has only one factor, 1. It can be expressed as 1×1 or 1x1x1…etc. Any prime factorization of a composite number is unique and so does not include 1 otherwise any composite number could have any number of factors, we could just extend it indefinitely by extending 1.

If we look at (3(7) we see it can be further broken down into (3(1)(1(7). Numbers like these two can be considered the shared counterpart to primes. Yet to have it, it is not a prime factorization. Yet, (2(2)(11(1) could be a prime factorization, since I have a single 2 and 1 but I also have two prime factors. Here then 1 is considered to be a prime number. Non-prime factorization just opens the shared number up and allows for non-finite representation. Yet, so does prime factorization!

So a prime in this system might be represented as (1(p) or (p(1) where p is a prime in the higher system. Then (p(q) where p,q are primes would be a different kind of number, we could call it a binary prime shared number.

If I do include this decomposition then (3(7)=(1(3)(1(7) =(1(21) and then if I have (10(10)=(2(2)(7(3)=(7(3)+(3(7)=(2(2)(21(1)=(42(2)=(22(22). So (10(10)=(22(22). This may lead to interesting mathematics, but if we allow it it means that the representation of a shared number is not finite. We need a finite representation otherwise the sum of the shared numbers is not conserved.

We can make a further restriction to only allow shared number decompositions where the sum of the two component numbers are equal over all possible decompositions. These means the sum of original numbers is conserved. Since we are sharing numbers, this makes sense that the amount we are sharing one way or another can vary but the sum of the numbers must remain the same.

This works out with a factorization where one of the factors is (2(2) as we can switch the two numbers in the other factor. This makes the sum the same. If we allow the sum to be different, there is no finite representation.

Here is an example: (2(2)=(3(1)

If we move by one up or down from (3(3), for example we can obtain (2(4) and ask is (2(4) okay? But (2(4)=(2(1)(1(4) (since 4 is composite this is an allowed decomposition)=(2(1)(1(2)(1(2)=(1(2)(1(2)(1(2)=(1(8) and 1+8=9 and not 6. We seek a prime decomposition which maintains addition of shared numbers.

Then for (16(16) we get (16(16)=(2(2)(p(q) and (13(3)=(11(5). No other binary numbers work as I can further decompose them and by switching lead to two different sums. Also I can’t decompose (13(3) and (11(5) further as this will not be prime factorization.

Think of (12(12)=(2(2)(6(6). So if we are at (6(6) I can move it to (7(5) and this works out because (7(5) has no prime factors of either 7 or 5. This is the way it will work. I must have for example (16(16)=(2(2)(11(5) the second factor has to be where both numbers have to be primes. This is then the definition of these shared even numbers, greater than or equal to (4(4), since they are conserving the sum of the numbers.

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.

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

This is a factorization, one is considered a prime number.

In general (2n(2n)=(2(2)(p(q).

So in the shared number system we need a new definition of prime factorization. Let’s look at some more examples.

(6(6)=(2(2)(4(2) but (4(2)=(4(1)(1(2) (4 is composite so it is okay to express the decomposition this way as the shared product is that of a composite)=(2(1)(2(1)(1(2) which does not work as I can break it down further into (6(6)=(2(2)(2(1)(2(1)(1(2)= (2(1)(2(1)(1(2) +(1(2)(1(2)(1(2)=(4(2)+(1(8)=(5(10). 5+10=15 not 12.

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(2)(2(1)(1(2). This can be further broken down to (2(2)(2(1)(1(2)+(2(2)(1(2)(1(2)=(4(4)+(2(8)=(6(12). This is not allowed. We find (8(8)=(5(3)+(3(5)=(2(2)*(3(5)=(2(2)*(5(3)=(10(6), and 10+6=16=8+8.

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

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, the left number moving up by one and the right number moving down by one.

We also can’t have multiple factors like (12(12)=(2(2)(2(2)(3(3) because I can have then (12(12)=(2(2)(6(2)(1(3) and then (12(12)=(2(2)(2(6)(1(3)=(2(2)(2(18) and it won’t add up. Having multiple factors again opens the shared number up for a non-finite representation. There must be some creation of (12(12) which has only finite representation. It must exist since I need to have 12. There must be a stable way of defining it. You see, if we allow (12(12)=(4(36) I can further extend (4(36) indefinitely.

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.

You see as we now have (4(4), (6(6), (8(8),…. we can now go on to create the usual natural numbers. The picture below, would now be in reverse. We would replace (1(1) by 1 and (2(2) by 2, etc.

Divide (4(4) by (2(2) to get (2(2) and divide (2(2) again by (2(2) to get (1(1). The we can go forward by dividing (6(6) by (2(2) to get (3(3) and (8(8) by (2(2) to get (4(4) etc. Then all this sequence can lead to the natural numbers. I can replace (1(1) by 1, (2(2) by 2, (3(3) by 3 etc.

This also 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).

Also, since we include 1 as a prime in some cases, it can be seen that there must always be at least two of these different representations of any even shared number greater than (4(4). So if we have one of them being (p(1) another one must be (r(s) where p,r,s are primes. For example with (10(10) we have (5(5) and (7(3). Since once I form the number 10, I need at least two of these to feedback to the other, original number line. See the picture below. We might also have more than two representations, In which case there are extra added dimensions.

Then there must be at least two of these decompositions for each even shared number. We might have one (p(1) but then I have another one (r(s). 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.