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.
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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.
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That should ALSO significantly shorten finding more and many primes to use in the RSA systems.
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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.
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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.
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Here’s the Addendum showing how all of this works.
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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.
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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.
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Addenda:
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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.
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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.

One thought on “The Magic of the Prime Multiples; Insights into Goldbach’s Conjecture

  1. Pingback: How to Find Primes Anywhere in the Number lLine, Fast & Efficiently, No Matter How Large  | La Chanson Sans Fin

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