Is Goldbach’s Conjecture True And/or False, Conditionally?

By Herb Wiggins, M.D.; Clinical Neurosciences; Discoverer/Creator of the Comparison Process/CP Theory/Model; 14 Mar. 2014

Copyright © 2018

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Essentially, we have to explore the primes to find the patterns & information to understand why Goldbach’s is proven, conditionally, as true, and is also not true, conditionally. We must then add the information to show this to be the case, logically.

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As stated before in the NP Not = P article, we must add information to most problems to solve those. Given the amount of Shannon’s information theory quantity of info then NP cannot be = to P. Therefore, in order to solve the hard NP problem of Goldbach’s Conjecture, that the even numbers can all be written as the sums of 2 primes, we must find/add information which can be used to solve that problem.

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And here those are, brought to us by the higher logics of the comparison process, Least energy and empirical, experimental math, the (futures of mathematics) to solve the problems of complex system understandings, as well.

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We must first look at the capabilities AND limits of our logics, and then we make and find some good answers, plural, NOT a single answer. Logic is not complete. Godel’s Incompleteness Theorem showed that important truth,from 80 years ago. Empirically, the EM number line of colours is NOT complete, shown by “The Structure of Colour Vision”.

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First of all, is 1 a prime? Goldbach clearly thought so. & then by solid, empirical, mathematical evidence showed that the even numbers were the sum of primes, & stated, going too far, that they ALL were. However, it’s proven empirically, experimentally, practically up to very high numbers. But not forever, and that will be shown soon, & simply, and why it’s the case.

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But is 5 + 7 = 12 true? Yes. Is 11 + 3 =14, as is 7 + 7 = 14 true? Yes. And the sum of all of these true statements of two primes added together to create the even number line, up to a very high number, are All true, too. The sum and collection of true statements are just as true as the truth of each statement. Thus, to very high numbers, Goldbach’s is true!!

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How DO WE know that 1 is not a prime? And that’s an axiom, an assumption, and how this is the case is based upon the inadequacies and inconsistencies, somewhat arbitrary of Number theory. Goldbach had no problem with 1 being a prime. If 1 is not a prime, then what is it? What’s a Non elephant? Do we not know? How can we? Calling something NOT something does NOT tell us what it is.

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1 is the relationship between all of the numbers in the number line, both above and below a specific number. The number 1 has a relationship, comparison process value, & therefore. It creates the number line, all the odd, even numbers and the primes, and all of the prime multiples, as well.

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If 1 is a prime, then Goldbach’s is easily proven true. We add 1 to each prime and we get a lot of even numbers. We add early on 1 + 1, and we get two. we then add 1 + 2 and we get three, we add 1 plus 3, and we get four, then 6, then 8, 10, and so forth.

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But we note these things. Does adding a prime to another prime create an odd number? Not at all. Adding two odd numbers, except for the case of 1 & zero, can create only even numbers, but empirically, not with odd numbers. Nor can adding any odd number to any other odd number create anything but an even number. That’s basic number theory. Adding a prime, except for the case of 2, which is a special case only works for the 2nd of prime twins to create another prime. and that’s because, also, adding an even number to an odd always creates an odd number. So ONLY by adding THREE primes can we get the odd numbers. & the primes.

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Now, IFF 1 is not a prime, then what is it? Or is math in part, the study of all not elephants, as Ulam’s koan stated? And that’s the problem, logic doesn’t work for complex systems. It’s not all A and Not A. & Godel’s Proof showed and Hofstadter affirmed that deep truth, “This statement is false” or the many permutations of it, “This sentence is not the case”, creates a real problem in logics. Everything is either all black or white, that is, white or not white. & that is empirically false. What of the unlimited shades of greys? What of the colors, AND the combinations of colours AND the shades of grays? Or DayGlo? The number line of EM spectra, is thus quite as incomplete, as the math is, by Godel’s Proof and empirically.

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Thus we have shown that Goldbach’s Conjecture depends upon an unworthy instrument for “proof” in mathematics. A or Not A demanded by logic in math, is not the case, empirically. That model does NOT always work in the real world. The false dichotomy is yet another way in which logic defeats and eats itself, like the Worm of Ouroboros (or the Procrustean Bed of Logics!) thus destroying itself. It’s all not “either/or, nor true or false, but the unlimited shades of greys, or colours, and their combinations. & what of our visual system? Comparing that to the EM line shows the limits of the EM linear, math lines, What of the combinations of all possible colors? What of Brown and reddish blues? Not on the EM spectrum. The empirical logic of the EM spectrum/structure is NOT complete. Godel’s incompleteness theorem, yet again. “This statement is not likely to be true”, is the problem. Limits to tools, and their capabilities, yet again. Structural limits to logics.

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Now we must consider negative numbers, to show, once again, these tough number theory problems. Let us take addition in positive numbers. When we add two positive numbers together, we get a sum of even numbers, not quite just like the basis of Goldbach’s Conjecture deals with. And when we start with 2, we count up to 8, and find that we must count up by 6. So we make a table to memorize the relationships of 2 and 8, being 2 + 6 = 8 (equally true, 6 + 2 = 8). And for all the other additions, as well, as that table efficiently (thermodynamics) short cuts “counting up” every time we add 6 to 2 to get 8. It saves us much time and energy. Long Term Memory does that, BTW.

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Then comes the trouble. We then find that the sum of 2 4’s, or 4 2’s is Eight. And that 8 minus 6 or counting backwards, and down, is the same, 2. Thus the generation, source of subtraction. Thus 2 + 6 is 8, or 6 + 2 is 8, empirically by counting. & subtraction is counting down, essentially simplified counting. Thus counting is valid evidence of the truths of the addition tables, is not? And the subtractions tables, is not? And that the relationships of all of the numbers as compared to all of the rest of the numbers can be expressed by counting, the baseline. Then the more efficient, the additions and subtraction tables as well. & thus we do generate arithmetic by comparison processing, which shows the relationships of the numbers to all other numbers, by simple arithmetic, is NOT?

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An hierarchy of counting, then adding, then subtraction; and then the next level, multiplication, and division and then long division, etc. 1’s 10’s, 100’s, 1000′, 10K’s, 100K’s and millions etc. Hierarchical all.

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That’s how arithmetic is created, generated and comes about. Very simply.

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Now what happens when we add the positive 2 4 times? We get 8. Thus the multiplication tables are validated and PROVEN by this means. A positive number plus a positive number is a positive number. And an odd number plus odd numbers is an even number. and an even number plus an even number is an even number. So we have that consistency. & an odd plus even number is odd.

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Now, comes the trouble. Negative numbers. If we add positive numbers to positive numbers, we, by convention, an axiom, get positive numbers. But if we add an odd number such as -2 to a -2 and up to -8, by the same subtraction numbers, we get the same additions using odd numbers as we do even numbers, do we not? That is the case, and the odd negative numbers plus odd negative numbers give even odds. And the odd negative numbers plus the even’s give odd numbers. and the even negative number plus even negative numbers, give even negative numbers. This is the rule.

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But what happens with multiplication is the trouble with the negative again. (& what of zero? (Oops.)

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So if we multiply an odd number by an odd number we get a positive number. The addition is consistent but the Multiplications are NOT! Using the same conventions, an odd, negative number plus a negative number should be a negative number, is not? But that’s the problem. & the whole of the imaginary number line is the case. Oops.

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Now if we try to factor positive numbers, we have not any troubles, But how do we factor large negative numbers such as -12? We cannot!!!! Because the thing doesn’t work that way, by convention. The Axiomatic problem with number theory is in the way of it. For -12 is it -4 times -3? Nope it’s +12. is it -2 -6, nope. Is it minus 4 times +3, to give -12? Or the other way around 4 times minus 3?. Hopelessly complicated and confusing. Something’s not right with the Number Theory because we cannot multiply negative numbers in any sensible way, NOR factorize them at all. Oops.

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And, what’s more we can add negative odd primes to odd negative primes and get negative even numbers, too!! BUT NOT factor them. There’s a problem here, and it’s what causes the trouble with Goldbach’s. IFF 1 is not a prime, then Goldbach’s arises & trouble with proving it. IF Goldbach’s 1 is prime is the case, Goldbach’s is trivially true.

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So here we are. 5 plus 7 =12? Are 5 and 7 primes? Yes. If Goldbach’s is true, it’s because that’s valid. What of 3 plus 3 being 6? what of 1 + 1 being 2? Not valid due to 1 not being a prime. More Non-elephants we see. So what is the number 1? We are not told.

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So, if we must therefore conclude that 1 is not a prime, Goldbach’s is a problem, because it does NOT allow 1, 2 or 3 to arise. so Goldbach’s Must be true, conditionally, for all even numbers >/= 4. Otherwise, it’s not true. Thus for what we see, Goldbach’s is true with that condition, if 1 is not prime, or a not elephant. Goldbach’s is conditionally the case, AKA true. But not quite, which opens the door for more objections to Goldbach’s being true.

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Then we know that since 7 plus 11 give us 18, and 13 + 5 is 18, and 11 + 7 also 18, that Goldbach’s is true. And thus for the rest of the even numbers. If each of the sums of the primes creates the even numbers, then the collection of those statements MUST be true as well, but conditionally, as 1 is not prime.

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Thus we have at least one new piece of information. Conditionally, Goldbach’s is true. Now the next step.

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Let us consider the prime number line and the Gaps in the Primes. Those spots are composed (by composites) of prime multiples and related numbers, which are not prime, and on either side of 120 by 7 units, from 113 to 127, there are NO primes. Now, what happens, assuming that the prime numbers (all but 2 being odd!!) create the positive even numbers when added?

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We have a LOT fewer prime sums, because of that gap in the primes, with which to create the even numbers. Each of the even numbers above 20 have a LOT of prime pairs that when added create those even numbers, NOT just one. So if we add, assuming that 117, 119, and 123 are Prime, for instance, we have a lot more prime sums which create the even numbers. So increasing the numbers of primes, increases the sum of primes for each even number. But since we have a great many primes which can do this, we ask this question. What if the primes thin out? At what point does Goldach’s become false? Here is the solution.

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If we, for instance removed all but 4-5 of the primes in the 100 centad to 200, what happens to the number of prime sums to create the even numbers? They stop doing so!!!. Thus we can prove that if the primes are reduced in number past a certain point, Goldbach’s which was formerly true conditionally above 3, can fail. But does that happen? And the answer is yes!!! Because eliminating the primes in the centad of 100, to create the even numbers does that. Whereas adding more primes increases the prime sums for each even number.

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And that’s the case, and must be. Because we know as the primes begin to fall, with vastly higher number line, we see the inverse log function showing us few, and fewer primes per centad, i. e., groups of 100 numbers such as 101 to 200, and so forth. & we know that the primes must eventually fall to about 0 – 4 primes/centad (and then even less), when the numbers get very large, and this can be calculated. And Proven to be the case. In the regions of 200 there are many primes, upwards of 15 or so. By 500-600 those begin to decline. By 12,000 more so being around 10 or so. & this can be computed out by a special function. Thus we know that the primes begin to decline regularly against an asymptotic limit. We never get to the last prime, but they become increasingly rare as the numbers get into the millions of digits. that is a proven fact, and thus yet another of the pieces of information falls into place to solve Goldbach’s.

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This is the rate of prime density drop off.

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lim Pn/n =1/ln (n) when n goes to infinity.

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About 1/2 way through this article:

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We’re up against an exponential barrier, asymptotic limits of the primes. Hmm, quite like Cee, 0 Kelvin, the Uncertainty Principle, and that there are NO perfect heat engines; nor in information theory, perfect descriptions, either for the same reasons. Have treated expon bars in many instances on the blog, too, and how those occur and what creates those to some extent.

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In those cases then, just as we see by ridding the number line of the primes in the 100’s, we cannot generate certain positive integers, either. Thus Goldbach’s fails with very large primes, because there are so few of them. But the problem is that calculating those huge numbers is beyond us, & thus the empirical, mathematical truth that Goldbach’s MUST, logically, mathematical fail with very high primes, as they are rarer, and rarer, falling down to 1-2 per centad, is not? And somewhere down the number line at about twice the size of those fewer primes centads, will be found the even numbers not amenable to creation by adding two primes.

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And that is the solution to Goldbach’s. It’s both conditionally true, but eventually fails. Logic can’t do this requiring either/or. The same is true of Riemann’s Hypothesis about the Zeta function, BTW. Thus the likely Trinary solution of Goldbach’s contributes to the next solution of Riemann’s.

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Thus, Goldbach’s is ONLY conditionally true, but NOT universally true, by yet another instance. Goldbach’s is not the whole case. It’s true if p is >/= 4 or higher, but NOT true for 2. And when the primes get rare, and this can be empirically in time proven, then Goldbach’s at some point must fail, is not? And we know this simply by removing most of the Primes from the 100’s centad and see the sums of primes drop to zero for some even numbers. & we know that by increasing the number of primes also increases the number of prime sums to create even numbers, as well.

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Thus, Goldbach’s is conditionally true, from >/=4 up to very large numbers, but not universally true for the number line at very high numbers.

QED

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That’s the epistemological power of the new complex system, comparison process logics, and how to create creativity. Add information of the right kind and problems get solved. Find that by efficient sorting methods, using comparison process & pattern recognition creating new information, least energy, structure/function (S/F) methods plus the unlimited methods created by the same.

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But rather, there is a Great Race coming resting upon the empirical solution to where Goldbach’s becomes not true, because even numbers will be found very far down the number line, which cannot be created by adding two “known” primes. This is the inevitable outcome of the facts that the sum of two primes will become no longer possible when the primes fall to about 1-4 primes per 100 number centad, & thus unable to support Goldbach’s.

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Because that is in the millions of digits size of numbers, by the inverse log function which shows that. It will be most easily found where a LOT of even numbers in a group, well downstream in the number line, will not be summable by any 2 primes. The first groups of even numbers, failing Goldbach’s, will be far, far upstream from that easier to find even numbers not summable from two primes.

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That will take HUGE amounts of processing power, sorting systems, and efficient programs to find. So the hardware, software and so forth will have to be developed to find out where that goes on. Quantum computers are highly efficient sorters. Those can solve this problem of the empirical test of Goldbach’s failing at very large numbers. Where Goldbach’s conjecture becomes not the case any more. Those empirical findings will cinch the final proof for the understating that Goldbach’s is both conditionally false for 2, then true to very many high millions of digits, but then becomes not the case, well up into the number line.

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And as the saying ago, now The Great Race is on AND, “The Game is Afoot!!!!”

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