The Magic of the Prime Multiples; Insights into Goldbach’s Conjecture

The Magic of Prime Multiples/Quartets; & Insights into Goldbach’s Conjecture
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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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“Almost anything which jogs of us out of our current abstractions is a good thing.”  –Alfred Whitehead
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“If you would discuss philosophy with me, first Define your Terms.” –Voltaire
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Gauss’ Razor:  Mathematics must be constrained by the practical uses of mathematics. AND by those findings which create greater understanding of  how mathematics works.
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Least Energy Rules
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“Understanding comes from finding the Relationships among Events.”  Albert Einstein, “Physics and Reality”, 1936
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The Future of mathematics is Experimental, Empirical mathematics relating to complex systems.
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The fatal, empirical flaw of most Deductive logic is the False Dichotomy.   (Either and only; All White or all Black. But, What of the unlimited greys?)
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“Calling the Universe Non-linear is like calling Biology the study of all Non-elephants.” —Stanislas Ulam
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“Simplify, simplify, simplify”  —Henry David, Thoreau.
“Efficiencies drive the Markets” —Adam Smith, the “Wealth of Nations”
Least energy, Least energy, Least Energy Rules.
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The essential basis of creativity depends upon complex system factors. Pattern Recognition is one high level basis of understanding. This in turn is based upon comparison processes which create information, and by doing so create creativity and understanding. By understanding high level information creation based upon how information is both created verbally (description) and by measuring (empirical mathematics, both thus invoking Einstein’s Relativity epistemology), the requirements of creativity become much, much more clear. This work is an active part of of a new model of cognitive neuroscience, and shows the power of the new understanding of the origins of information, viz. description (Verbal) and Data (counting, measuring), and how those are organized hierarchically & can create unlimited creativity and understanding.
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 Recently there was a found a method which quickly and efficiently sorts through the number line to find the primes. Using the new concept of Prime Multiples (PM), this can be successfully done without limit and very quickly, too. Prime multiples (PM’s) are different from the usual numbers called composites, because they are a simplified number line consisting of n >/= 11, of the numbers ending in -1, -3, -7 & -9. Those 10 quartets make up 40 of the numbers in each centad (100 numbers), and have special properties. All of the primes after  5 end in odd numbers, of the -1, -3, -7, -9 types. So that method clears the number line of all the not primes.
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A Prime Multiple(PM), in its specialized definition, means one of those numbers in the number line which is composed of at least 2 primes mulitplied together, or more, ending in -1, -3, -5, -7, or -9. Those PM’s cannot by definition be primes. This is how those arise & are identified.
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First of all, the number line is generated by adding 1 + 1+ 1 +1 …. to each number, which creates the number line & creates the many kinds of numbers, including the primes. Primes Cannot be generated or created in any other way (except as below, Addendum), altho they can be ID’d and shown to be primes. The primes cannot be created except by using that process of  1 + 1 + 1, etc. Primes (except as below, see Addendum), cannot be generated either by arithmetic. Very quickly we see that the numbers ending 2, 4, 5, 6, 8, and -0’s can be eliminated from the prime lines after 7, very easily using the prime quartet simplification. Simplification of the number line is a key to sorting out and finding the primes. As Thoreau so stated.
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By multiplying the known primes after the 4 numbers ending as above, have been removed easily by using the quartets, because all primes .>/= 11 end in 1, 3, 7, & 9, this becomes possible to reduce the number line to about 26% (73.3333…% of numbers removed) by this means alone.
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Now, because primes Cannot be generated except by 1 + 1 + 1…., added to each number, if 1 is considered prime, then the solution to the Goldbach conjecture that all numbers can be expressed by the sum of two primes becomes a trivial solution.
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However, if 1 is NOT a prime, which is a definition, but not an absolute, & an axiom, then Goldbach’s conjecture becomes much harder to solve. Very likely, Only if we consider 1 not to be prime, then does Goldbach’s problem surface. Goldbach’s is not true then for the 1st even number, 2, so Goldbach’s is NOT completely true. But is true for the rest of the number line to a very high number. That problem will be addressed in the article on proving that Goldbach’s is both true, and not true, conditionally, and broadly.
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However, using the prime multiples (PM) definition it becomes very clear that if we eliminate the PM’s from the number line, then the primes can be simply found.  Because all numbers of the 1, 3, 7, 9, quartets but for the primes are then clearly prime multiples (PM’s). All primes (p) are created by 1 + 1 + 1, but then they begin to repeat at 2p, 3p, 4p, and so forth without limit. But the initial primes are NOT the sums of primes, but of 1’s. Thus all of the numbers but for primes & but for 2 are all odd numbers, & the even numbers can be the sums of two primes, or more. But not quite.
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Each prime starts out as a number not heretofore seen in the number line. As the adding of numbers continues, then the primes begin to repeat as the prime multiples of 2p, 3p, 4p, 5p, etc. By filtering out those repetitions to ONLY the odd numbered PM’s  using the 1, 3, 7, 9, quartets, we can much more easily show how the number line is composed of new primes, and the much larger part, esp. in the very high numbers of 1 billions or more, the PM’s, in which complexities are found, by exclusion of the PM’s, showing the primes. Exclusion is NOT a mathematical function. But based upon a process where numbers not meeting the standard of PM’s are found, then removed leaving the rest, i.e., the primes.
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And we can empirically find two primes which by adding together create the even number line. This is largely true for  the higher primes, that is >/= 11. 7 is 5 + 2, but not all primes for which that can be seen to be true but for the twin primes.
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Because those prime lines are additive, then all but the primes are necessarily prime multiples (multiplication being an easy process to reduce, logically, to additions), and thus Goldbach’s conjuncture is the case, but for all primes & odd numbers. Ergo, Goldbach’s conjecture, to say it another way, is that if the primes & odd numbers are eliminated, we are only left with PM’s & THOSE are the sums of primes, all of them. So Goldbach’s is conditionally, in a limited form, restricted by PM’s, the case. IOW, as in sorting the primes, simplify the problem down, and then solve it.
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This at once shows the power of the Prime Multiples definition, not as composites, but because it clearly can sort out very quickly all the lower primes, altho is laborious, but NTL true. The works of Dugas/O’Connor and Juhani Sipila are very supportive of the finding of primes to very high numbers, (1 billions) using the composites removal method from the number line.
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For instance, there is with the quartets method the clear fact, that all primes create prime multiples of the 1 class numbers ending & being  1 X 1, 3 X 7, and 9 X 9. The -3 class  are 1 X 3, and only -7 X -9; the 7 class, 1 X 7, & 3 X 9; and the -9’s by 1 X 1, and 3 X 7, and 9 X 9. That’s the whole story. And that is why the PM method works. It generates all odd numbers by multiplication only, & removing the PM’s from the quartet line of 40 numbers per centad (1oo numbers group) leaves the primes.
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Further, the powers of the primes all progress with the squares ending ALL in -1’s and -9’s. & then they by a simple, repeating arithmetic/multiplication to ending in the 3’s, 7’s, 1, and 9’s again with each power as each rises.  This is inviolate.
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With the PM’s a number series can be made of an arithmetic series beginning with the quartets with 7 X 7, being 49. Adding 4X7  (4p) to that, then 2p  to that sum, and then repeating the simple two part wheel ( +4p, +2p, +4p, etc.) to  find, create, generate, all the PM’s of 7 to unlimited values. The same pattern is true for all primes beginning with 7, 11, 13, 17, 19, etc. And by using this very simple method, the primes are sorted out very quickly by this simple processor which removes the basic PM numbers from the quartets, leaving only the primes, very efficiently.  See Addenda for the examples of how this sorts out the primes uniquely and completely from the number lines.
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By sorting out ONLY the PM3’s by a simple pattern, it saves many calculations on the odd PM’s and increases the efficiency of this method many fold.
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Because it’s a series of p squared plus +2p , + 4p, plus 2p,  + 4p, etc., the entire line of each prime as its PM’s can be easily found & removed. As the average is 3p, then it proceeds 3 times faster than the E Sieve. And further, because 75% of the number line has been removed, 12 times faster!!!  At least.
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Further, because the occurrence of primes are the case to very large numbers, always greater than 1, the number of primes/centad also means that those primes do not need to be sorted out, lowering the numbers to be processed by 8-16 per 100, thus removing 83.333 to 88% of the number line!!! We need only sort out the PM’s and the primes are left. This speeds up prime sorting again by about 10-15%. Esp. efficiently in the first 10K of 100 centads in the number line, where the primes are greatest in number.
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Then we begin to see the magic. By removing the PM3’s from the quartet number line to 77, we have the primes remaining, up to 77. By then finding the 2p and 4p PM7’s of the primes, those PM’s (77, 91, 119, 133….)  start very quickly being removed, again leaving the primes to 143, where in the PM11’s start to drop out, sorting out more primes. By then generating the PM13 series after 169 (13 Sq.) we find the primes to 191 and then begin removing the PM13’s up to 323, where the PM17’s start continuing the PM removals. This finds a HUGE number of primes, by only using 7, 11, 13!!  The ratio of advantage is by the PM3’s being removed, by the pair of PM3’s in the quartets; which used at once after the 1, 3, 7, 9, quartets are created, & we sort out 61 primes by using only 3 primes up to 323!!! A very good ratio of efficiency.
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 This can be done with the entire simplified number line using calculators or manually. Using this arithmetic progression of the PM’s, it’s possible to sort out the primes into the 15K-20K+ list of numbers on paper, without using a computer or calculators. & it does Not miss one prime at all, if the accounting is exactly done. It’s that simple and efficient.
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When THAT is done, the relationships of the PM’s to each other, creates factorization solutions to the larger problem of factorization. Many numbers are divisible by 2, 3 and 5’s (5 and -0 ending numbers).. But when those are taken out Simple additions of the primes creates the factors to the PM’s embedded in that simple number line. We find interactions, the same numbers with many prime factors, implicit in the series, for instance among the PM 7’s, 11’s, 13’s, 17’s, etc., as below in the Addendum.
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The problems of factorization occur because the PM’s, and in all of their unlimited products. Factoring out numbers by 2, 3, 5, and -0 is trivial. But that leaves the hard, NP problem of the odd PM’s. And when those are all known, and this method creates them, factorization can be VERY fast.
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For instance, 7 X 4 is 49, add 4p or 28 and we get 77, 2 primes, creating the 77, 7 and 11. Adding 2p to that gives us 91, 7 X 13, the next prime; and then 28 again gives us 119 (7X17), then 14 again 133 (7X19), and finally at 143, the 11 PM elimination series then begins. Thus by using ONLY the PM7, after clearing out the PM’s by simple inspection, we have found the first THIRTY primes after 7,  Having cleared out the prime multiples from the number line simply, easily and cleanly.  Dividing the PM’s by the generating +2p, alternating with +4p reveals the Exact primes line, as well, up to about 113, when it becomes more complex.
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At that point in each PM times 113, the PM’s line become the same as the numbers added to the primes. It’s not all  primes, mostly primes early on, but also the PM’s. As single primes  are only used once, but the PM’s then begin to dominate. As in the series of all odd numbers, minus the PM3’s. That’s the line of multiples which then begins. and this creates ALL of the multiples of the primes (PM’s), their squares and powers to p exp. x. This simple arithmetic system  of easy additions, then finds all of the PM’s which can be used to do the hard factorization problems. This means, that factorization becomes a LOT easier, once we remove the even PM’s, and the -5’s/-0’s. THAT’s the hard problem of factorization, the odd numbered PM’s. Solving THAT problem solves and speeds up factorization methods. And that is known, but now it’s seen why due to the quartet systems making the PM’s stand out.
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And note that the 7 X 7 is not prime, the next is 77 or 7 X 11, and then next 7 X 13, 91, & then 7 X17, 119, etc.;  the sums divided by the prime series being used, show the primes as well!!!. That is the first Rem1 quartet of all 2 pairs of primes, 11, 13, 17, & 19. So the prime numbers are sorted out to about 7 X 113, in EACH of the primes starting with PM7 sorting. It’s like magic. & nothing is missed. Not squares, not cubes, not powers of p, not anything. Adding the 2p & +4p’s, this series creates the prime line as well in factorizing numbers to the many higher primes still being used!!!  Those lists of intersecting PM’s provide the raw data for very fast factorization of the number lines. Esp. among the PM5’s which are regularly generated by the system. And in addition those odd numbers of the primes are duplicated EXACTLY for each prime used from 7, 1, 13, 17, 19, to the highest primes known.
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And thus the overlaps between the PM’s lines show the factorization of the number line. Thus, when we eliminate the trivial 2, 3, 4, 5, 6, 8 and -0’s ending numbers, we are left with the bare bones possible prime numbers. Thus the primes find the primes, at least twice, over. By specific exclusion from removing the PM’s from the number line, as well as, by dividing EACH of the PM’s shows the same primes. That is nothing short of Magic, even if we ignore that removing PM3’s by inspection ALSO finds the primes up to 77, of itself. & by using only the PM7 line, the simple adding of the PM’s starting with 7, 11, etc., ALL generates the primes by dividing the PM series by the prime driving that series to 113!!! After that the PM’s are being seen interspersed with the primes, altho there is an easy way round that, as well.  The method has already sorted  out the PM’s by having already ID’d them as NOT primes. Thus leaving the primes in the PM times the series of odd primes and PM’s. And these are the same, exact PM factors for EACH PM series. Thus when we get to 101, we are using the same prime and PM list to create the continuing primes sorting, as well as prime generating sequences. IOW, 101 X 101 (101Sq),  plus 202, plus 404, +202, etc. leaves 101X 103, 101X107, 101X109, 101X 113; 101 X 127 after a few PM’s in the prime gap about 120.
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Thus, once again speeding up the prime sorting by another 30-60%, depending upon how many intersecting PM’s are being created. Adding up the p sq. to the 2p and 4p highlights the primes, too!!!! Thus by a simple process of the 2P and 4p additions, we generate numbers, which when divided by the primes series number base prime,  find the primes by a simple division!!!
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So we can use TWO methods to find the primes, the first creating by arithmetic progression the PM’s, which when removed from the odd quartets leave the primes. And the Second, which is composed of the odd number line created by multiplying the base primes against that. For instance, it all starts out with the prime times the prime, at 7, 49, and then 7 X11, &X 13, 7X 17, and on up till be get to 49, where the prime line breaks, and then up to 71, 73, and then the break at 77, 91, etc. being PM, and then 79, 83, 89, 101, 103, 107, 109, and 113. After that more PM’s are seen, and it gets very much less the primes, only. But as we KNOW what the PM’s are, when those are removed from the Prime times odd number line, bereft of PM3, it gives once again, diving the numbers by the prime series of the 7, 11, 13, etc., the prime line!!!
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This is astonishing. It gives the primes twice, the first time by excluding the PM’s and the second time by dividing the prime multiples by the odd number prime driving that line, less the PM’s, as well. So the one system backs up the other for sorting out the primes.
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Curiously, to state it another way by adding the squares of the primes to the 2p and 4p sums, then dividing many of those, shows the primes.
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Thus, for instance, the number line
7X:  113, 119, 125 127,  (quartet)131, 133, 137, 139, we see the 131 quartet primes and the sole PM 133, which is 7 times 19!!! This continues by removing the PM’s to create the Primes!!!
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And that works for each PM series, too. We are creating the primes by addition of the series of p sq. +2p, +4p, =2p, etc. Bit more complicated than that, but is the case.
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So we get to 7X (quartet) 191, 193, 197, and 199. The PM series create the primes by arithmetic, and faultlessly and without limits.  And by subtraction of the PM’s by a simple pattern, this simple division creates the primes, but ONLY if the PM’s are known. And as the primes are well known before the P sq. plus 2p, +4p gets going, that method also shows the primes, as a b/u event to detect errors!!!
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That’s another of the Magics of the PM methods. Instead of counting up, 1+1 +1 +1, we add up the PM series to find the primes as well. How this occurs is a great mystery, but seen easily. We are finding the primes, by arithmetic progressions, and in every case, too. & then generating the primes by using semi-primes, two primes multiplied together. THAT is the heretofore NOT seen PM effect/phenomenon. And with each number it does this faultlessly AFTER it has found the primes by exclusion.
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So if we removed the PM’s we can find the primes by exclusion once again. Magical, very astonishing and not expected.
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Let/s look at this in the high detail with the PM method uniquely gives us.
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The prime 7 series of PM’s is essentially 7X7=49, then 7×11=77 by 4p addition, and then by 2p addition, 7×13 =91, and then 7X19, 119, and 7X23 is 161.
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So essentially what we are doing is taking out the PM3’s by removing those from the quartet system. But we already KNOW the primes up to 143. So we can use those to remove ever more quickly as we have the prime series of primes & those numbers. This takes the PM’s out of the prime factors created by the arithmetic progression and saves us many steps in terms of checking.
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For instance in Prime series of 19, that starts at 19 squared, and runs up to by 23, 29, 31, 37, 41, 43, 47, etc. through 113 by simple adding.
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So we have 19X 23 which is 4p or 361 + 76 is 437. We ignored the 21 number as it’s PM3. and we have the 49 to use, and have eliminated the 5’s too. so we have this
19X19 = 361
19X23 = 437
19X25 = 475 which we skip
add 19X 29, skipping the 27 as it’s PM3, and get 76 plus that to give
19X29 = 541 well past 23 sq. at 529; and then the 29 times 25 occurs at  575 (5 times 115), and then next is 23X29 at 667, etc.
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So the odd number line proceeds by removing by a simple not mathematical process (saving lots of time) all of those numbers divisible by 3, and only a few by 25 , & 5 of its powers and multiples, which can be ignored. But acts as a sum to add the 2p, or 23X2 or 46 to and we find that overlap which lets us simplify the additions by removing the repetitions, easily. thus increasing the speed of prime exclusions.
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At higher numbers, this becomes more complex, but the overlaps among the PM’s are easily removed by a simple pattern observation.
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Let’s next discuss the PM3 rules by which the primes can be found very quickly. It’s simple, but very tedious to cast out 3’s using the usual adding of digits. But using the quartets methods, we see at once, a pattern of PM3 pairs in the cast out 3  zero remainder, c/o 3 9, and c/o 2, repeating quartets. When zero, the -3 and -9 ending numbers are all PM3’s, because zero plus 3 or 9 is a PM3. When the remainder number is 1, all 4 of the quartet can be primes. When those are 2, then only the -3 and -7 numbers can be primes. Thus we see the repeating primes of the forms, ending in -1, -7, and then in c/o 3 remainder 2 quartets, the 3 and 9. & following this up thru the known primes in the first 200, we see this pattern all the time, too. the 23 and 29, the 31 and 37, the 41, 43, 47, skipping the PM 7 sq. 49, and then the 53 and 59, the 61 and 67, and the 71, 73, % 79, the 77 being 7 X 11. Then the 83, and 89, and the lonely 97. Thus do the PM3’s determine many of the patterns of the primes. And ONLY when the 2 quartets transition to the zero, do we find twin primes, but not withIN the 0, 2 quartets alone, by themselves. And there are possible double twin primes in the c/o3 1 quartets all the time. And then patterns in those patterns.
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Then we begin to realize that in the PM lines we see the route to easier factorization. Even numbers and 5’s, 0’s and 3’s are very easily eliminated by simple inspections. The REAL problems arise with the odd PM’s. But this method shows how THAT can be solved as well. because in the trivial, Even, 3, 5 and -0 ending numbers, the biggest problem is the -1, -3, -7, & -9 ending factors. The PM’s method neatly solves this without limit!!! It creates the factorizations!!! & thus provides new insights in how to apply those findings to more efficiently factor the number line.
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& those can be used as cross checks to the additions, as well. The PM5’s are seen at about 1% and declining with higher numbers. The PM5’s arise as a result of the arithmetic additions of the PM’s. By simply dividing the numbers ending in 5 arising with the PM5 series, we find ever more factors of the PM’s. Those also create the factorizations not including the PM3 5’s multiples. which is just more of the PM magic, too.
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The PM5’s occur at 10 and alternating 20 times the Prime 7 (70 & 140), 11 (110 & 220), 13 (130 & 260), etc., values. so that can also be used to check the accuracy of the PM additions series. If a mistake has been made, then those 10 and 20 times PM5 patterns will not be seen. Thus it’s yet 1 more cross sums checking for the accuracy of the p sq. +2p, +4p series, etc. More magic!!!
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And it also highlights by this simplification the nature of the Zeta function, as well. BTW, except for the very high primes, Riemann’s Hypothesis is very likely correct. His series generation shows the PM’s, and leaves by elimination, the Primes. Implicitly and specifically in the Zeta function, therefore is the p sq. plus 2p, +4p, plus 2p, +4p series. THAT’s what has been missed. and because the PM system of quartets is the case, that is, each sum of 2p and then 4p repeated is mathematically true. SO then is the Zeta function and Riemann’s Hypothesis true. Because the multiplication (by 2p and 4p additions) of the two primes is true, thus the series collection of all the sums, is ALSO true. And Riemann’s is then proven the case. The PM’s show this proof. It’s true, because summing up numbers is true in each case. and the divisions are true, as well, which show the primes, and the PM’s. Whether this also holds in the gigantic numbers can be eventually, experimentally shown.
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Thus do we both provide solid evidence of Goldbach’s Conjecture being conditional, and Riemann’s being likely true by the same PM method.
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Further, When we realize that the prime gaps show us what’s going on, then we reach greater understanding.
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Let’s reconsider once more this important point. If by finding the PM’s we remove those from the 1, 3, 7, 9, quartets, we find the primes, and we do, without a miss, then we know what the primes are. & can use them to extend the primes lines. But if we find that in the PM’s themselves, the product of the single Primes times the odd number line divested of the PM,s also shows the primes, then we have an interesting insight.
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We can remove the PM’s from the factors of the number lines and then directly find the primes, as well.
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The Prime gap of 113 to 127 shows us what occurs, in full details and specifically, in a nutshell, with ALL prime gaps. That’s the first one, and it’s 7 below and 7 above 120, thus 120 +/- 7. That is NO accident. 120 is divisible by huge numbers of primes & PM’s. This again shows us the prime factorization solutions, as well.
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This is very interesting, too. Because when we sum up 2 primes to create the entire number line of even numbers, those numbers show yet again the Rem PM3 zero, 1, & 2  recurrences, the repeating 3 quartets of PM3 driven processes. The 30’s, 90’s and the 60’s, too. The largest numbers of two primes sums creating an even number rises as the even numbers rise. AND there is a 6 repeating series, which quickly becomes yet another 30 series. Such as 8 unique prime sums for 48 becomes TWELVE two prime sums creating 120!!! & with 360, very, very many prime sums. The most up to that number in fact. So the quartet methods shows how that is found & why, too. The PM3’s shape and determine the primes.
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Again, the prime gaps signify the high rate of interference of the numbers which creates the first major prime gap from 113 to 127. And which being extensively created by PM’s is also the case with most all the rest of the prime gaps. & the two prime sums creating the even numbers. These patterns of the sum of two primes giving the entire even number line (as far as can be tested with today’s computer tech) is a very deep insight into why Goldbach’s conjecture is true, and provable. But NOT quite!!!
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And the most of the totals of the two primes sums are also often multiples of 3 and 6 and & 9 & 10.
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When we look at the double pairs which occur with 11, 13, 17, & 19, we add 3 times 30, and get the next double pairs in the c/o 3 Rem 1 quartets. That is 101, 103, 107, and 109. The next is 191, 193, 197, and 199, again, 90 units away. and we find this occurring time and again. This is not explained, but needs to be. The quartets produce a great deal of new information about the primes, which expands our understanding of mathematics, significantly.
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821, 823, 827 and 829 are 191 minus 812, 630 numbers separated, again, 90 times 7. That 90 keeps recurring with the double pairs of primes in the c/o 3 Rem 1 quartets!!! Why?
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In addition, when we sum up the number of primes with respect to the cast out 3, Rem 0, 1, and 2 quartets we find early on that most of the primes occur in the Rem1 quartets. With higher numbers this tends to go back to a balance among the primes in each quartet with respect to how many PM3 pairs are found in each centad. But frankly, because the Rem1 quartet can have more primes than the other two, it’s remarkable that often it has 50-60% of the primes in the quartets in each centad, early on. Altho as stated above, it tends to even out with higher numbers. It’s still clear that if the Rem 1 quartets are checked using primality method such as AKS or ECPP, those still with very high numbers of digits are like to have the most primes in the Rem1 quartets. So with very high numbers where only zero to 4 primes are found per centad, the money is STILL in the Rem1 quartets for finding primes. That’s new information and data about the number line and primes.
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And in the twin prime finding from  Rem2 -9 to Rem0, -1, Those patterns of primes should save lots of time in finding the primes..
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ALL prime gaps are the results of the PM’s creating composites and PM’s in succession which are NOT primes. And the prime sequences of +2p or +4p, such as often seen can be thought of as this. Prime gaps are the reinforcement of the sequential numbers of PM’s, which eliminate using the recurring PM3 structure of the quartets, the number line, leaving the primes.
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In addition, the succession of primes often seen only separated by 2 and 4 such as in 13 to 17, and occ. 6 are the same, and can be conceived of as not reinforcement of the PM’s but break thru the PM blocking patterns in the expression of the primes. Thus the prime gaps are self-reinforcement by the PM’s and groupings of the PM’s; and the lengthy prime sequences seen, are the interference patterns. This is what’s going on, and once again shows the power of the quartet system. This is best seen in the lower centads where primes are still very common. Less often after 20K or so. See Addendum below to visualize those patterns.
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The primes are the cast of the statue, not the statue itself, which are created by the PM’s. And when the statue (PM’s) are removed, the primes remain. This visualizes the relationships of the primes and the PM’s. The PM’s create transient prime patterns because of this, but those do not persist. Only the PM patterns persist and are real and the same throughout the number line. Viz., prime squared plus alternating with + 2p and +4p series to unlimited numbers.
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We can go into the PM3 ordering of the quartets at length, but this also shows the power of the quartet method of 1, 3, 7, 9, too. because invariably all the -1, -3, -7, -9  2 pairs of primes, are seen in ONLY the quartet which is cast out 3 Rem1. In remainder 0 or 2 has a pair of c/o PM3;.  in Cast out 3’s.  The 1 remainder has no PM3’s. And summing up the primes in each centad, we see quickly that the  Rem1 quartets is at least 50-60% of the primes, by itself. the Rem2, and Zero being usually less than 1/2 of the primes. Altho at times it can be a bit more, or even, but not usually.  The 100’s of primes known easily can be separated into the c/o 3, of the  0, 1,  2  groups, and shows that clear cut pattern. The primes LIKE the Rem1 quartet because it can hold up to 4 of them and in any combo from none to 4. 2 twins, pairs, separated by 4, between the -3 and -7.  & many triplets primes, as well. And we see most often early one the -1,-7 ending primes in the Rem0 groups because the -3 and -9 are PM3’s and cast out;  & the similar c/o -1, -7’s, which leaves the -3, and -9 numbers to be primes. That’s why the pattern is there!!! The PM3’s drive the prime patterns.
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So if we want to quickly find primes, we focus on the Rem1 quartets. That has NOT been seen nor understood, either, before. and thus also shows the power of this system to detect the deeper patterns of the PM’s and primes.  Any pattern in the primes ALWAYS reflects the underlying PM3 imposed upon the far more complex PM pattern. And as this can vary, creates pseudo patterns of the primes, but which never, ever recur or easily. The patterns of the primes are always incomplete, and appear random, but for the fact that they reflect the TRUE additive 2p plus 4p patterns which are dominant.
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& is we want to find the primes fasger, check the Not PM3 numbers in the Rem0 and Rem2 quartets.
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The 90’s rule shows this. in the succession of 2 pairs of primes  of 11, 13, 17, & 19, are where they are. The next is 101, 103, 107 & 109. Which is 3 X 30 numbers away. Then we find the next double paired primes in the 191, 193, 197, and 199 again, Rem 1 quartets. 3X30 = 90, again away from the last primes pair. Even at 630 quartets later, we find yet another set of 1, 3, 7, 9 pairs. That shows the power of organizing of the PM3 patterns in the quartets and the primes. This does not hold up, however, as written above. It always breaks down. There are NO steady patterns to the primes, but those which occ. occur as a PM pattern, driving it by exclusions of the PM’s..
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In addition, to close, the number line at ANY place (even to 100’s 1000’s or millions of digits) can be investigated by constructing the 1, 3, 7, 9, quartets, and then using the simple arithmetic progression ridding the number line, most efficiently of upwards of 99+% of the numbers!!!. And thus concentrates the primes. Thusly allowing faster methods, such as the ECPP related methods to find the primes at 10, 100, 1000 and even 10K times faster.  Instead of dividing those numbers of 2, 3, 4, 5…. into the number line, we use the bare bones quartet primes method of PM’s to simplify,  simplify,  simplify the number line. Thus gaining HUGE time savings, by least energy sorting. Many 1000’s of times faster. If the testing for primacy of numbers can be  reduced by 100 fold, or even 1000’s fold. If the calculations of primacy in a large series of many 100’s of digits of numbers takes say 2 months, it’s reduced by the PM exclusions method to a number of hours, not even days, rather. It’s very much faster than doing an E-sieve number reduction up to 150 numbers, say, because by using PM61, the the number line is already reduced by over 95%.
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So let us take a series of 350 digit numbers and we want to find the primes in those. We find, successively the PM3 pairs in the quartets & remove those. Then sequentially remove the PM7, 11, 13, upwards of PM41 to 61, which removes 95% of the not primes in the number lines
with only several primes. If this is expanded further by the PM’s series of primes it very quickly rises.
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That means the number line has been simplified by 1000 times, leaving the primes to be found VERY quickly by AKS or ECPP of the remaining numbers after the PM sieve has processed them in only a very short time. The method Concentrates the primes, first by simplifying down by 60% the number line, then upwards of 73.333% and then with each PM7, PM11, PM13, etc. ever more so, and quickly rising.
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This means that in calculations to find series of primes for encoding credit cards &  bank accounts, a great many can be found very, quickly, with 1000’s of parts upwards of 10K portion of the number line removed by these arithmetic, simple, PM series sieving.
.
This is so much faster than simply doing an Eratosthenes sieve to remove a large part of the number, line, say up to 157, because it rises very much more quickly to clear out and concentrate the primes.
.
That should ALSO significantly shorten finding more and many primes to use in the RSA systems.
.
For number lines in the 100’s of digits, this cuts the primality numbers to be tested by 1000’s fold, for instance. Thus for calculations to find a high primes normally taking many weeks, it only takes several hours or a few days to do the same work.
.
Simplify, simplify, simplify the number line to find the primes. The primality tests cannot do that very easily nor as efficiently as the PM’s method does by simple arithmetic of the 2p + 4p additions. No complex maths. Just an accelerated E-sieve to at least 12 fold or more.
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There are other, almost unlimited numbers of patterns which can be easily detected and uniquely so by using this method of PM sorting. And that is the promise & Magic of the Prime Multiples and the unlimited methods which that can create a better understanding number theory via the (VIP) very important primes.  & also in a practical sense, sort out the primes, much, much faster, to the chagrin of RSA systems. As we have often suspected, but can’t always prove.
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And there are MANY other patterns found here, which time/space do not allow to be discussed and explained. Unlimited new methods regarding understanding the primes, very likely.
.
Here’s the Addendum showing how all of this works.
.
Note the patterns of the primes in Rem1, 2, and zero quartets, which are seen again and again. Note the twin primes being ONLY seen in the move from the Rem2 -9, to the initial _1 in the Rem0 quartets. Note the twin primes being found in the Rem1 quartets consistently, too, and in every variation, as well.
.
Note the Prime sequences also seen,  but rarely mentioned in the p plus 2 and P +4 occurrences, and WHY those occur, and why ONLY two pairs of primes can be seen & no more, because the PM3’s make this happen. This new information is very useful in understanding more about the primes, and satisfies Gauss’ Razor by not only being useful in finding primes, but also in understanding the 30, 90, and PM3’s multiples patterns being seen.
.
Addenda:
.
The 1, 3, 7, 9, series will be shown thru the 500’s centad, and then the corresponding primes generator series examples will be shown Adjacent to it.
.
1. Pr means prime;
2. ~ (tilde)3 means NOT prime of a PM3 series sorted out
3. Prime Multiples will be shown by ~( 7 X 73 for instance, thus sorting out the PM’s and showing the remaining Primes.
4. Starts and ends of longer prime sequences twin primes and those Primes separated by only 2 and 4 units shown by *****
5. Rem0, -1, or -2 means one of 3 repeating quartets marked by cast out 3 series
.
Rem 2              PM7 Series
501 ~3             511 +28 (7 X73
503  Prime       539 +14 (7 X77
507 -3              553 +28 (7 X79
509 Pr              581 +14 (7 X83
—-                    595 PM5, +28  (7 X85 (5X17 Overlap
                         623 +14 (7 X89  End PM7 series for centads
.
Rem 0
511 ~(7X 71
513 ~3
517 ~(11X 47
519 ~3
——-                PM11 series start
Rem 1              517 +22 (11 X47
521 Pr              539 +44 (11 X49 PM7 overlap, above
523  Pr              583 +22 (11 X53
527 ~( 17X31    605 +44 (11 X55 or 5X121 (End of PM11 series
529 ~( 23 sq.)
23 series starts here  +46 to 575 (23 X25 (5 X5)
—-
Rem2
531~3
533 ~(13X41         PM13 series starts
537~3                    533 +26 (13 X41
539 ~(97 x77(11×7)
—-
Rem0                     559 +52 (13 X43
541 Pr                    611 +26 (13 X47
543 ~3                  etc of PM17, PM19, etc.
547 Pr
549 ~3
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Note same prime progressions, exactly for each successive prime series   that is Exactly the same and extends up to unlimited numbers. Thus, if the 7x  primes and PM’s is known up to the prime #, 601, the same line exactly can   be used to compute the PM’s of every prime up to 601. No point in doing it again & again..
————–
Rem1
551 ~(19X 29
553 ~ (7X79
557 Pr
559 ~(13 X 43 (+52
Rem2
561 ~3
563 Pr
567 ~3
569 Pr
——–
Rem0
571 Pr
573~3
577  Pr
579~3
—-
Rem1
581 ~(7 X83
583 ~(11 X53
587 Pr
589 ~(19 X31
—-
Rem2
591 ~3
593 Pr  *******
597 ~3
599 Pr  ** ( quartet Rem2 to Rem0 twin primes, only twin primes outside of Rem1 quartets
———————–
.
Rem0
601 Pr **
603 ~3
607 Pr **
609 ~3
—-
Rem1
611 ~(13 X47)
613 Pr  **
617 Pr  **
619 Pr ******* end of prime sequence
Etc.

Table of Contents

Updated 17 Nov. 2018

1. The Comparison Process, Introduction, Pt. 1
https://jochesh00.wordpress.com/2014/02/14/le-chanson-sans-fin-the-comparison-process-introduction/?relatedposts_hit=1&relatedposts_origin=22&relatedposts_position=0

2. The Comparison Process, Introduction, Pt. 2
https://jochesh00.wordpress.com/2014/02/14/le-chanson-sans-fin-the-comparison-process-pt-2/?relatedposts_hit=1&relatedposts_origin=3&relatedposts_position=1

3. The Comparison Process, Introduction, Pt. 3
https://jochesh00.wordpress.com/2014/02/15/le-chanson-sans-fin-the-comparison-process-pt-3/?relatedposts_hit=1&relatedposts_origin=7&relatedposts_position=0

3A.. Extensions & Applications, parts 1 & 2.

https://jochesh00.wordpress.com/2016/05/17/extensions-applications-pts-1-2/

4. The Comparison Process, The Explananda 1
https://jochesh00.wordpress.com/2014/02/28/the-comparison-process-explananda-pt-1/

5. The Comparison Process, The Explananda 2
https://jochesh00.wordpress.com/2014/02/28/the-comparison-process-explananda-pt-2/

6. The Comparison Process, The Explananda 3
https://jochesh00.wordpress.com/2014/03/04/comparison-process-explananda-pt-3/?relatedposts_hit=1&relatedposts_origin=17&relatedposts_position=1

7. The Comparison Process, The Explananda 4
https://jochesh00.wordpress.com/2014/03/15/the-comparison-process-comp-explananda-4/?relatedposts_hit=1&relatedposts_origin=38&relatedposts_position=0

8. The Comparison Process, The Explananda 5: Cosmology
https://jochesh00.wordpress.com/2014/03/15/cosmology-and-the-comparison-process-comp-explananda-5/

9. AI and the Comparison Process
https://jochesh00.wordpress.com/2014/03/20/artificial-intelligence-ai-and-the-comparison-process-comp/

10. Optical and Sensory Illusions, Creativity and the Comparison Process (COMP)
https://jochesh00.wordpress.com/2014/03/06/opticalsensory-illusions-creativity-the-comp/

11. The Emotional Continuum: Exploring Emotions with the Comparison Process
https://jochesh00.wordpress.com/2014/04/02/the-emotional-continuum-exploring-emotions/

12. Depths within Depths: the Nested Great Mysteries
https://jochesh00.wordpress.com/2014/04/14/depths-within-depths-the-nested-great-mysteries/

13. Language/Math, Description/Measurement, Least Energy Principle and AI
https://jochesh00.wordpress.com/2014/04/09/languagemath-descriptionmeasurement-least-energy-principle-and-ai/

14. The Continua, Yin/Yang, Dualities; Creativity and Prediction
https://jochesh00.wordpress.com/2014/04/21/the-continua-yinyang-dualities-creativity-and-prediction/

15. Empirical Introspection and the Comparison Process
https://jochesh00.wordpress.com/2014/04/24/81/

16. The Spark of Life and the Soul of Wit
https://jochesh00.wordpress.com/2014/04/30/the-spark-of-life-and-the-soul-of-wit/

17. The Praxis: Use of Cortical Evoked Responses (CER), functional MRI (fMRI), Magnetic Electroencephalography (MEG), and Magnetic Stimulation of brain (MagStim) to investigate recognition, creativity and the Comparison Process

https://jochesh00.wordpress.com/2014/05/16/the-praxis/

18. A Field Trip into the Mind

https://jochesh00.wordpress.com/2014/05/21/106/

19. Complex Systems, Boundary Events and Hierarchies

https://jochesh00.wordpress.com/2014/06/11/complex-systems-boundary-events-and-hierarchies/

20. The Relativity of the Cortex: The Mind/Brain Interface

https://jochesh00.wordpress.com/2014/07/02/the-relativity-of-the-cortex-the-mindbrain-interface/

21. How to Cure Diabetes (AODM type 2)
https://jochesh00.wordpress.com/2014/07/18/how-to-cure-diabetes-aodm-2/

22. Dealing with Sociopaths, Terrorists and Riots

https://jochesh00.wordpress.com/2014/08/12/dealing-with-sociopaths-terrorists-and-riots/

23. Beyond the Absolute: The Limits to Knowledge

https://jochesh00.wordpress.com/2014/09/03/beyond-the-absolute-limits-to-knowledge/

24  Imaging the Conscience.

https://jochesh00.wordpress.com/2014/10/20/imaging-the-conscience/

25. The Comparison Process: Creativity, and Linguistics. Analyzing a Movie

https://jochesh00.wordpress.com/2015/03/24/comparison-process-creativity-and-linguistics-analyzing-a-movie/

26. A Mother’s Wisdom

https://jochesh00.wordpress.com/2015/06/03/a-mothers-wisdom/

27. The Fox and the Hedgehog

https://jochesh00.wordpress.com/2015/06/19/the-fox-the-hedgehog/

28. Sequoias, Parkinson’s and Space Sickness.

https://jochesh00.wordpress.com/2015/07/17/sequoias-parkinsons-and-space-sickness/

29. Evolution, growth, & Development: A Deeper Understanding.

https://jochesh00.wordpress.com/2015/09/01/evolution-growth-development-a-deeper-understanding/

30. Explanandum 6: Understanding Complex Systems

https://jochesh00.wordpress.com/2015/09/08/explandum-6-understanding-complex-systems/

31. The Promised Land of the Undiscovered Country: Towards Universal Understanding

https://jochesh00.wordpress.com/2015/09/28/the-promised-land-of-the-undiscovered-country-towards-universal-understanding-2/

32. The Power of Proliferation

https://jochesh00.wordpress.com/2015/10/02/the-power-of-proliferation/

33. A Field Trip into our Understanding

https://jochesh00.wordpress.com/2015/11/03/a-field-trip-into-our-understanding/

34.  Extensions & applications: Pts. 1 & 2.

https://jochesh00.wordpress.com/2016/05/17/extensions-applications-pts-1-2/

(35. A Hierarchical Turing Test for General AI, this was deleted after being posted, and it’s not known how it occurred.)

https://jochesh00.wordpress.com/2016/05/17/extensions-applications-pts-1-2/

35. The Structure of Color Vision

https://jochesh00.wordpress.com/2016/06/11/the-structure-of-color-vision/

36. La Chanson Sans Fin:   Table of Contents

https://jochesh00.wordpress.com/2015/09/28/le-chanson-sans-fin-table-of-contents-2/

37. The Structure of Color Vision

https://jochesh00.wordpress.com/2016/06/16/the-structure-of-color-vision-2/

38. Stabilities, Repetitions, and Confirmability

https://jochesh00.wordpress.com/2016/06/30/stabilities-repetitions-confirmability/

39. The Balanced Brain

https://jochesh00.wordpress.com/2016/07/08/the-balanced-brain/

40. The Limits to Linear Thinking & Methods

https://jochesh00.wordpress.com/2016/07/10/the-limits-to-linear-thinking-methods/

41. Melding Cognitive Neuroscience & Behaviorism

https://jochesh00.wordpress.com/2016/11/19/melding-cognitive-neuroscience-behaviorism/

42. An Hierarchical Turing Test for AI

https://jochesh00.wordpress.com/2016/12/02/an-hierarchical-turing-test-for-ai/

43.  Do Neutron Stars develop into White Dwarfs by Mass Loss?https://jochesh00.wordpress.com/2017/02/08/do-neutron-stars-develop-into-white-dwarfs-by-mass-loss/

44. An Infinity of Flavors ?                             https://jochesh00.wordpress.com/2017/02/16/an-infinity-of-flavors/

45. The Origin of Infomration & Understanding; and the Wellsprings of Creativity

https://jochesh00.wordpress.com/2017/04/01/origins-of-information-understanding/

46. The Complex System of the Second Law of Thermodynamics

https://jochesh00.wordpress.com/2017/04/22/the-complex-system-of-the-second-law-of-thermodynamics/

47. How Physicians Create New Information

https://jochesh00.wordpress.com/2017/05/01/how-physicians-create-new-information/

48. An Hierarchical Turing Test for AI

https://jochesh00.wordpress.com/2017/05/20/an-hierarchical-turing-test-for-ai-2/

49. The Neuroscience of Problem Solving

https://jochesh00.wordpress.com/2017/05/27/the-neuroscience-of-problem-solving/

50. A Standard Method to Understand Neurochemistry’s Complexities

https://jochesh00.wordpress.com/2017/05/30/a-standard-method-to-understand-neurochemistrys-complexities/

51. Problem Solving for Self Driving Cars: a Model.

https://jochesh00.wordpress.com/2017/06/10/problem-solving-for-self-driving-cars-a-model/

52. A Trio of Relationships and Connections

https://jochesh00.wordpress.com/2017/08/04/a-trio-of-relationships-connections/

53: Einstein’s Great Subtleties:  Einstein’s Edge

https://wordpress.com/post/jochesh00.wordpress.com/583

54. The Problem of Solving P not Equal to NP

https://jochesh00.wordpress.com/2018/04/28/the-problem-of-solving-p-not-equal-to-np/

55. How to Create a Blue Rose

https://jochesh00.wordpress.com/2018/06/02/how-to-create-a-blue-rose/

56. The Etymologies of Creativity

https://jochesh00.wordpress.com/2018/06/14/the-etymologies-creativity/

57.  A Basic Model of a Unifying System of Most All Knowledge

https://jochesh00.wordpress.com/2018/07/06/a-basic-model-of-a-unifying-system-of-most-all-knowledge/

58. Understanding Psych with S/F Brain Methods

https://jochesh00.wordpress.com/2018/07/11/understanding-psychology-with-s-f-methods/

59. The Wiggins Prime Sieve

https://jochesh00.wordpress.com/2018/08/02/the-wiggins-prime-sieve/

60. The Complex System of Love

https://jochesh00.wordpress.com/2018/08/22/the-complex-system-of-love/

61. The Limits of the Comparison Process

https://jochesh00.wordpress.com/2018/08/27/the-limits-of-comparison-processing/

62.  The Bees, Cortical Brain Structure, Einstein’s Brain, etc.

jochesh00.wordpress.com/2018/09/14/the-bees-cortical-brain-structures-einsteins-brain-the-flowers/

 

63. The Wiggins Prime Sieve, Version 3.

https://jochesh00.wordpress.com/2018/09/15/the-wiggins-prime-sieve-version-3/

64. The Prime Quartets Method

https://jochesh00.wordpress.com/2018/10/04/prime-quartets-method-capabilities-insights-sans-limits/

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

https://jochesh00.wordpress.com/2018/11/17/is-goldbachs-conjecture-true-and-or-false-conditionally/

66. The Magic of the Prime Multiples and Goldbach’s….

https://jochesh00.wordpress.com/2018/11/27/the-magic-of-the-prime-multiples-insights-into-goldbachs-conjecture/

67 The Wiggins Primes Sieve:  Cycles of 30’s in the Primes

https://jochesh00.wordpress.com/2018/12/17/the-wiggins-prime-sieve-cycles-of-30s-in-the-primes/

68. Winning at Solitaire, Basic Strategies

jochesh00.wordpress.com/2019/02/04/winning-at-solitaire-basic-strategies/

69, The Failures of Idealisms & Brain Hardwiring in the Sciences

jochesh00.wordpress.com/2019/04/04/the-failures-of-idealisms-brain-hardwiring-in-the-sciences/

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

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.
 .
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!!!!”