|Dec25-12, 04:49 AM||#1|
Some ideas concerning the Goldbach conjecture
Me and a friend, David Barrack, are non-mathematicians but we've been having fun lately with the Goldbach conjecture. I thought I'd share some of our tools with you guys, some of you might be interested in helping us to progress on this problem - that would be greatly appreciated. I apologize in advance if anything I say is not termed in the way you are used to, I am not a mathematician and I never really read about maths, except for Euclid’s book, so I might not be used to the terminology. I’ll try to make it as clear as possible.
The goal here is to frame the Goldbach conjecture in geometrical terms. The Goldbach conjecture states that every even number can be expressed as the sum of two primes. Say N is our even number, we draw two lines y = -x + N and y = x. Since N is even these two lines intersect on an integer point and from that point going to the left every integer coordinates on the y = -x + N line is a set of point that summed together give N (let's say they are "candidates" primes). Out of these points, about half of them are even and won't be prime, so we are mostly concerned about the odd pairs. For instance, for N = 16, the pairs (7, 9), (5, 11), (3, 13), (1, 15) are candidates (1 is excluded but we still keep it for later to generate our triangles and squares).
So if the Goldbach conjecture is false, then we should expect some day to find a N that is even and for which every odd coordinate points on the y = -x + N line is not a pair of primes. This has strong consequences for the odd coordinate points along the y = -x + N axis because it means that starting at the coordinate (N/2, N/2), making steps of 2√2 (because we ignore the steps of even pairs) along the y = -x + N line down to 1, I should find only odd coordinate points that have factors and since what makes a number prime is that it has no factors beside 1 and itself, then we should expect to find a set of factors between 1 and the number itself for each of these points, at least for one of the coordinates. What are the obligatory characteristics of those factors?
We know that they have to be integers, and somewhere between 1 and the coordinate. We also know that the factor is either unique and multiplying it by itself gives the number (for instance, 9 has 3 and 3), or there are 2 of those that multiplied together give the number (for instance, 391 has 23 and 17). Thus stating that every odd coordinate points along the y=-x + N line would have at least one of them not being prime has important implications.
We have illustrated the consequences of this using 3 methods.
The first one consists in staying within that space and just drawing the rational curves that correspond to every combination of numbers from 1 to N (in the illustrated version, we draw every line, even those that correspond to even coordinates). Those curves are illustrated in Figure 1 for one N. The equation is simply y = i/x for i from 1 to N. Here non-primes have some intersection with the grid (some integer coordinates other than 1 and i) and the primes are expected never to intersect with the grid of integer coordinates. There is an important property here. Increasing i = 1 to N, each of these curves has to be higher than the preceding for the same x. That’s because increasing the numerator in the fraction y = i/x can only increase a number for a given x. This also means that whenever one of these curves has an integer point, for any of the further curves as i increases, if the curve has an integer point, it has to be a different point, higher or to the right, of the last point. Figure 2 illustrates each integer points that are part of the curves y = i/x for i=1:N. I noticed that each of the point within the square (√N, √N) has to be part of one of the curves and only one. Same thing with the integer points along the border of the axes up to N shown on Figure 2. This is normal considering that the curves are always increasing, it can’t go back and touch a point 2 times. The interesting property of those curves is that they are rational, this means that whenever they intersect the grid of integers at anything else than 1 or the number itself, then the number is not prime.
Figure 3 shows an ellipse that is formed by the very same curves, just rotated and translated to the middle of the graph (N/2). This requires some more thinking but if any of you want to check these things, my impression is that there is often (maybe always, at least as far as I could test manually), one of the coordinates on the inside border of the ellipse that can be intersected by a line parallel to y = x such that one of the intersected point has X coordinate being the first prime, and the other point intersected has Y coordinate being the second prime.
The illustration compressed in the small (√N, √N) square is interesting but another way to put it is to look at the squares of the odd coordinates (Figure 4). So imagine we have N = 34, what if we look at the squares of the odd coordinates (33, 1), (31, 3), (29, 5) … Figure 4 illustrates this. In blue are the square of those sets of two coordinates where at least one coordinate is not prime, in red those where the two coordinates are primes. Now to each square, I added a rational function inside the square and whenever any of the points of those rational functions has an integer coordinate (a set of factors that multiply to the number), we added a green point. To be prime the number has to have only 1 and itself as factors, thus it has no other factors and no green points within the square. There is some thinking to do here, what does it imply to have factors (green points), what geometric consequences does it have on the other squares? I noted that since we only have odd coordinates, then the factors have to be odd too because we need odd factors to obtain an odd product. Thus if those numbers are all odds and all have factors, how does this constrain the smaller region? Would there be enough odds available to make up squares for every odd coordinates? Does this have consequences on the distribution of the early square and what would it mean if each of these squares were blue, if each of these squares had 1 factor? Would it still be possible that N is even and that N/2 is integer?
Finally, another way to put it is to go along each of the coordinate points, like for 16, it would be (1, 15), (3, 13)…. (Figure 5) and from there draw a rational curve that is simply translated to fit in a square along the coordinates, one going up (3), one going down (13). This leads to a lot of complexity, especially at the middle and if you want to work out the consequence of this system go ahead and have fun. For now I note that the way I translate those curves, (simply adding +1) makes them all intersect with even numbers on the upper and left axis. Also, because those are rational curves made from odd numbers only, their intersect with the grid have to be odd because any intersect with the grid is a set of factors and the factors of an odd number have to be odd. But here everything is translated + 1, so each of these curves actually intersects at even coordinates only. There are many links between each of the green dots that appear; first they come from intersecting a rational curve on factors so they are symmetric: there is either only one at the middle (3,3 for instance) or a pair of them (23, 17) for instance. Also I’m not a mathematician, correct me if I’m wrong, but when there’s only 1, it has to be a prime because the square of a prime is the only way to obtain the square of a prime through multiplying integers. The red lines kind of limits what can appear on each side because they are the intersect of the 0.5 coordinates of each rational curve drawn in this way, thus they have an odd Y and factors on one side are constrained to appear within that curves, because everything below this is non-integer.
I created a video that shows the factors being added to a grid in this way. You can have a look here (Video illustration). The video starts from 16 and does all even numbers up to 980. The green points on the video are integer coordinates of the rational functions (they are factors that make up for some odd numbers at the split, say (23, 9). When the numbers are primes, there are no green points beside at the axis where the rational curve intersects the origin. One can see a very interesting conic section behavior at N/2 as the video progresses. Somehow the addition of a new even N is constrained by the past history (you can see the factors just cycling to the left or to the right) and yet there is a +2 that’s adding to the middle. It would be important to find what is the equation that governs the evolution of the first factors at the border of N/2 as these might be crucially related to future Ns as we progress further in the evens. Why are they organized in a conic section so often and what would this look like if an even N did not have enough “new” primes that could be summed to that new N?
The goal of all this geometrical framing is basically to create some hypotheses about some N that would violate the Goldbach conjecture and we would be glad to hear what the community has to say about this. There are a lot of open questions here and it would be nice to see what you can get out of it, so I’m releasing the Matlab scripts that can be used to draw these graph and would appreciate if any of you finds the interest and time to look at this and come up with some geometrical constraints that make it impossible for an even N not to have primes that sum up to it. For now my intuition is that the Fermat theorem needs to be considered, the fact that when a N grows of 2 it either has a N/2 that’s odd (having an impact on the left) or one that’s even (having an impact on the right). The fact that square root of 2 is irrational and that adding 2 to a coordinate on our grid amounts to adding 2√2 to the diagonal axis on the graph. We also need to consider that on the diagonal line, running from 1,3,5,7… to N/2 through each odd steps corresponds to making steps of 2√2. What would happen to these squares if every step of 2√2 we make, we meet a set of odds that are not primes, i.e. that have green points on their rational curves ? One other thing to note is that all of these rational curves, wherever they are drawn, have a strict position in the axis system : they are √i away from number i, so they at least more than √2 away from any of their anchor point, which is an important constraint. Especially in our system where we care mostly about evens vs odds – a step of √2 is particularly important.
I wish everyone nice holidays and hope to read your comments and ideas about what kind of geometrical property we would expect to happen at N/2 that would constrain the addition of both new prime numbers and new even numbers. I’m pretty convinced geometry might be a productive way to do this. Here since everything in the video looks symmetric I like to take the geometrical problem from the middle (N/2) instead of starting at 0: why is it that starting at N/2 I will cross a pair of primes such that the rational curve on the left and on the right of the middle do not have any integer points ? What could we say of a series of odd sets 1:N that would not sum to N ?
Matlab scripts to generate those graphs are attached.
|Dec25-12, 08:41 AM||#2|
Here's a supplement figure, to show a more generalized idea of what I did in Figure 4, at least for when N/2 is odd. Basically, is there any series of odd numbers from 1 to N/2 that would produce what we see on this graph (black dots intersecting the rational curves of the square of the odd at integer odd values within the square) and that would do it for every and each odd number of the series, and in those cases where it doesn't do it, that then N-A would do it (intersect its own rational curve). I think that's a possible way of framing it.
Figure 6 :
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