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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The primes have some patterns, but those are clearly most all the reflections of the underlying Prime Multiples of 3’s(PM3’s) repetitions, as modified by the other PM’s.

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For instance, where there are twin primes those occur in two set patterns. The 2nd commonest is in the 3 PM3’s groups, which repeat every 30 numbers to unlimited numbers. Essentially, each quartet is 1 of 3 repeating PM3 forms.The first is where the -3 and –9 ending numbers are always PM3’s. This allows the -1 & -7 numbers to be primes. Thus there can be NO twin primes Within this PM3 quartet form. The next is the quartet form where there are NO PM3’s and that allows all 4 to be primes, or in any combo of 1, 3, 7, 9, too. The last in this repeating succession of 30, is the PM3 quartet form where the -1 & -7 numbers are PM3’s & the -3 & -9 can be primes. But there cannot be any Twin Primes in these PM3 quartets. This is an absolute (dare it be said!) invariable in the quartet system and is a rule.

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Thus, between those repeating pairs in the 1st and last quartets, in those no PM3’s can be primes. This creates the only possible twin primes ending between the 1st and 3rd PM3’s quartets, and must be -9 in the one quartet and then in the following quartet, the -1. & indeed those are seen all over the prime lists. Such as 59, and 61, or 149 and 151, without limits. Those twin primes are made possible and driven by the structure of the 2 quartets around PM3 patterns. And because those PM3 patterns are invariable and repeating, there are Twin primes of that kind without limits. This proves part of the twin prime conjecture, very clearly.

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The last half of the twin prime proof is based upon yet again a series of patterns. In the middle quartet there are NO PM3’s exclusions of the -1, -3, -7, -9 ending numbers. Thus all 4, or 1, 2, or 3 or none can be primes. & in ANY combos. We Also can see NO primes in that quartet, but rarely. And can see -1, or -3, or -7 or -9 only, each alone. Or -1 & -3 twin primes, or -7 & -9 twin primes, or the -3 gap and then -7 patterns. And curiously, if we see the quads, or triplets or twin primes in those patterns we know we have seen the PM3 free quartet once again.

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But there is this caveat. The PM3’s can present with a -1 & -7 pair of primes, As can the middle, PM3 Free quartet. And also the PM3 with an open -3 and -9, can also be seen. But never a -1 and -9 as that only can be seen in the PM3 free quartet. That pretty much exhausts all known possibilities. And conversely NO triples, or -1 & -3; or -3 and -7; or -7 & -9; or -1 & -19 pairs, can be seen in the PM3 two quartets. So it makes finding those very easily in the quartets. Those are the quartet Identifying methods which the PM3 driving patters create.

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At 11, 13, 17, 19, we see that ONLY 4 primes series are possible. Next come the 101, 103, 107, & 109 primes. And next come the 191, 193, 197, & 199 primes. This is no accident, and again, those ALL occur at 90 number intervals. This is PM3 driven, clearly. & we see the 41, 43, 47 prime triplets there, as well. & the 71 &73 twins, as well. As those possibilities exist without limits in the no PM3 middle group of repeating -1, -3, -7, & -9. thus those twin primes can be seen, as can the triples, without limits in the number lines. Thus the twin primes conjecture is proven by the simplifying and universal quartet model of the primes. This is empirical mathematics and creativity.

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But why, additionally are NO more than Four primes possible in the entire number line? Again, the PM3 driving pattern. Because the PM3 quartet form of PM3 of -3 & -9 precedes the middle quartet form all possible 4 primes in a row, and the next PM of of -1 & -7 ends the PM3 quartet of 4 possible primes, again, this PM3 constraint does NOT permit more than 4 primes in a row!!!

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For instance:

171 PM3

173 Prime

177 PM3

179 Twin prime

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181 Twin prime.

183 PM3

187 PM 11X17

189 PM3

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Then

191 Prime

193 “

197 “

199 “

Then

201 PM3

203 PM 7X29

207 PM3

209 PM11X19

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Thus there cannot ever be a twin prime involving the PM3 mid quartet with a -9 preceding it, or an PM3 free quartet ending with a 9. That is circumscribed by the nature of the PM3 quartet sequences, as well. A sort of lemma of the why there are not any primes longer than 4 in the mid PM3 quartet. It’s impossible to be seen. Thus where the twin primes of -9 and -1 are seen it’s Always after the last PM3 and next to the -1 of the two PM3 pair quartets.

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& that is not all, so in EVERY case, the prime twins, triples, & quartets can occur & recur without limits in the mid PM3 free series. And where those are seen at intervals of 30’s, always.

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For instance, 821, 823, 827, & 829, are at 630 (90 X 7!) plus 191, 193, and 197 & 199, the double twins. Again, a 3 multiplier and this continues without limits. The double pairs of primes always occur in a pattern, early on, which is 90N plus the initial 11, 13, 17, 19, double twin primes, as well. However, as the double twin primes in the no PM3 quartets rise in values, this breaks down, possibly due to the entrance of the higher prime number multiples, altho it appears occasionally, even then.

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There are the PM3 driven twin primes. And WHY they are there. It’s not an accident but a deep structure of the PM3’s seen in the prime lines, which would have been missed had not the quartet system shown these deep structures by simplifying down the number line to ONLY those bare bone numbers which can be primes. Simplification is least energy, and a founding part of creativity and understanding.

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& that’s largely what’s going on. The quartet of -1, -3, -7 & -9. show the prime patterns repeating throughout the number line and are very useful mathematical tools to understand how the primes appear and why, in a clear cut, structural, explanatory way.

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There is a lot more to these primes, however. In all cases where the PM3 quartets the one ends in a -9 prime, and then the next twin prime can start with -1. This is the pattern and seen everywhere, without limit. There are NO limits at first to Twin primes and triplets and quadruplets until much higher in the prime lines.

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But a very interesting event occurs as the numbers rise into the 100’s of digits. The number of primes falls to 0 to 5 per 100. And that means twin primes still can occur, but become rarer and rarer. Tho the line is unlimited, those twin primes can still be seen. The same is true for the triple and quadruples. This has broad implications, and proves as well, yet another aspect to the fall off in primes. Which will be addressed, later.

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And there is yet another pattern which is very clear cut and real with respect to the primes in these quartets, which bare bones, stark efficiency of these 3 forms. allows it to be seen. Because ONLY two open spaces exist in the two PM3 quartets, but FOUR potential primes are in the PM3 absent quartet, thus not surprisingly up to 1000, there are about 99 primes in the PM3 free quartet and about less than about 92 , totals in both PM3 quartets. The numbers of primes in the PM3 middle is more than twice as great as the total number of Primes in the Two PM3 quartets. That also does not change, either. but likely gets rarer and rarer as the primes naturally, empirically diminish in numbers per centad of quartets.

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One further repeating pattern found is that the prime quadruples begin at 11, 13, 17, 19, then 90 units away at 101, thru 109, then 90 units away at 191 thru 199. Then a quadruple is not seen again until 821 thru 829. Which is 630 and thus 90 X7. Why this should be so is obvious, but again, will let the astute figure it out for themselves.

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Altho the 90’s intervals break down & get rarer, this is still seen from time to time in the fewer and much higher prime quartets in the PM3 free quartet in the middle of either PM3 pairs quartets.

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There are yet many other interesting patterns, but will let the astute accountants and mathematicians find those for themselves using the quartets method.

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Recommended testing it for more useful patterns, which are clearly, very definitely there and invariable throughout the number line.

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& that also allows a very, very fast removal of PM3’s thus yielding the other primes 11, 13,17, 19, etc. sequences of the primes, too.

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Commend those interested in understanding much more about the primes to use this method for more study of the primes. Undoubtedly there are many are hidden patterns winch will emerge if looked for by pattern recognition methods of creativity in mathematics, as detailed in “La Chanson San Fin”, which was used to create, find & employ and apply, the prime quartet system.

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The prime quartet method shows the deeper prime structures created by the PM3’s , and other prime multiples as well, such as the prime gaps. & shows that ALL of those are created by Prime Multiple reinforcement using this method. The prime gaps are prime multiples created, in every case, the first of which at 113 to 127 is very, very clear cut and demonstrative as well. & the PM3 groups make that very much more common than without.

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More on Twin primes

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There will always be twin primes but as primes become rare at the rate of 4/100, those will become much less common, and necessarily fewer repeats of twin primes. This is a matter of empirical fact. Where those twin primes max out will be in the earliest primes, as the patterns show, too. As the primes become rarer, and the 4 primes in a row become rarer, yet still, occasionally, following the 90+ rule, the number of twin primes will fall. But likely will stop when the primes/100 fall at very high numbers to 0-1 prime/100, by common sense. So it will be a low probability of twin primes, but not an absolute, either.

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Thus the maximum number of twin primes will occur early in the number line, and be reduced as the number line gets very large. That will be found to follow a pattern of 30’s and 90’s, as well. The max number of prime triplets, quads and so forth will also decline. & occur further and further apart in the number line. The number of primes can vary, even at 1 billion by as much as none/100 to as many as 12/100, as shown in the series of 50,847,000 primes found by Dugas/O’Connor. It should not be hard to find a logical reason for this, or proof, either.

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Now that it’s known that the twin primes are driven by a recurring pattern pf PM3 quartets.

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There is one last, huge implication of all of this. That the number of primes will continue asymptotically to near zero, but, like light speed, and 0 Kelvin, NEVER reach that. That exponential, or asymptotic limit is also clear. It will NEVER reach zero, but will get so close as no matter. Comparison process rules create the asymptotes and exponential barriers. & this outcome, like there are no perfect heat engines in TD, or the Heisenberg UP, is part and parcel of the comparison process’ nature, too. Yet another of unlimited asymptotes/exponentials, such as in the S-curves of growth, as well. Unbounded and created by the CP. Such are unlimited using CP methods.

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Thus the twin primes conjectures are easily, uniquely and clearly solved by the prime quartets methods, which greatly simplify down the number line to ONLY those ending in -1, -3, -7, -9 and thus shows the PM3 restrictions which creates that same prime line, too. As well as the PM3 patterns which highly restrict where the primes can fall.

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& it can further describe and improve our understanding of the twin primes, too. Those can ONLY be created in 2 places. The two possible -1, & 3, or -7 & -9 of the middle PM3 free quartets, and between the last, and the first PM3’s pairs quartets sequences.

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To repeat, much more can be found by using the quartets model. It’s essentially unlimited, except by an exponential barrier, which being asymptotic, will in time give the diminishing returns outcome, just like any S-curve or ANY method which is least energy efficient, for that matter.

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