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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It’s become clear that the periodic cast out PM3 patterns of the repeating Rem0, Rem1, and Rem2 quartets drives the primes. When we show those primes we get very interesting patterns.

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Rem0: 2, 3, 5, 7,

Rem1: 11, 13, 17, 19

Rem2: 23, 29

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0: 31, 37

1: 41, 43, 47, (but for 49, exactly 30 plus 11, 13, 17.

2. 53, 59, (exactly 30 plus 23, and 29

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0: 61, 67, (exactly 30 plus 31 and 37, PM3 pattern, excluding -3 & -9

1: 71, 73, 79, (exactly 30 plus previous 41, 43,

2: 83, 89, (excluding PM3 at -1, -7, exactly 30 plus 53, 59, etc. throughout the entire number line, again.

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0: 97, only

1: 101, 103, 107, 109, (exactly 90 plus the 11, 13, 17, 19, as above.

2: 113 (1st major prime gap around 120 +/-7 to 127.

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This accounts for the primes being driven by the PM3 patterns, and why the number 30 keeps recurring, even in the largest two prime sums of the numbers. Where the 3, 6, and 9 are seen in 18 sq., 324, we see a very large number of prime sums to create the even number 324, same as 120, 240, 360 and other numbers as well. This is a repeating 30’s pattern in the prime sums underlying Goldbach’s Conjecture, and it relevant to understanding Goldbach’s.

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For this reason the p squ. plus 2p, +4p is seen as another form of that repetition, where 2p plus 4p for 7 is 14 and 28, giving a sum of 42, which is 6 times 7.

and for 11, 22 and 66 which is 6 X 11.

And for the prime 7 PM5 repeats which are 70 units and 140 units apart, again, 210, or 7 times 30.

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& for 11, 110, plus 220 repeats of PM5’s which gives 330, which is 11 times 30, again.

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Thus the natural repeats of the primes are within that number line wide pattern of repeating 30 quartets of 0, 1, 2 remainders.

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This is also true why the 6n +/- 1 is used. And why the Prichard method uses mod 60 to increase the PM’s sorting, but very inefficiently compared to the 1, 3, 7, 9, quartet, a far simpler method.

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In addition, when we look at the number of primes, we see that declining patterns of the numbers of primes are VERY clear and smoother than when using the 10’s & 100 units to see how many primes there are.

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For instance: add up primes here, from each 0-90, 91 to 180, etc.

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We see a much smoother drop off than when using a base 10 estimator for the numbers of primes, & when we use a 30 units method. it’s even smoother.

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The 30’s unit provides a far, far more accurate way to estimate the numbers of primes by 30’s and 90’s segments, even to high values, than does the base 10 or log method!!! 30’s is the natural pattern of the primes.

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# of primes

0 – 90 24

91 – 180 16

181 – 270 17

271 – 360 16

361(19sq)

to 450 15

451 – 540 12

541 – 630 15

631 – 720 14

720 – 810 12

811 – 900 15

901 – 990 12

991 – 1080 14

1081-1170 12

1171-1260 13

etc.

.Note the extraordinary numbers of 3’s in the sums of primes in the list. This is not an accident.

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& it’s even smoother when taken by the prime sums of 30 cycles of Rem-0, -1, -2 quartets of total primes.

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That is the actual, empirical progression of the decline of the primes, too. & smoothed out even more with the quartets.method.

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Progression of primes by 3 quartets of Rem0, 1, & 3 method

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# primes

1 – 30 10 361- 390 6

31 – 60 7 391- 420 4

61 – 90 7 421- 450 6

91 – 120 6 451- 480 5

121- 150 5 481- 510 5

151- 180 7 511- 540 2 Prime Gap

181- 210 5 541- 570 5

211- 240 6 571- 600 5

241- 270 5 601- 630 5

271- 300 5 631- 660 6

301- 330 5 661- 690 4

331- 360 6 691- 720 4

721- 750 4

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So why has this not been seen or reported? Logic is not very creative of itself. Empirical, experimental methods show the patterns of events in existence, which are detected those by repeating events. & then math tries to catch up, but cannot, logically, because as Godel showed, in his “Incompleteness Theorem”, that logic cannot describe it all. Which is, once more, why Riemann’s Hypothesis & Goldbach’s Conjecture are so hard to solve. The evidence & patterns which explain and potentially validate what’s going on, very likely, are in the numbers, not logic.

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Totals of primes sorted by Rem0, 1, and 2 quartets in 21K to 21,900, or 30 30’s.

Primes in quartets, 21K

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Rem0 Rem1 Rem2 Total primes per 90 numbers/quartet

21K to 4 4 3 11

21090 to 2 5 3 10

21180 to 3 4 1 8

21270 to 1 6 1 8

21360 to 2 4 2 8

21450 to 3 6 1 10

21540 to 3 7 1 11

21630 to 1 4 2 7

21720 to 4 4 2 10

21810 to 3 4 3 10

21900

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Totals 26 48 19 93

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The last ca. 50 primes before 1 B, from Dugas & O’Connor,

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from 999999,001 to 1 B.

Totals

Rem0 12

Rem1 23

Rem2 7

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Total 43

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And note that the -3, -9 primes in the Rem0 groups also exceeds that in the Rem2’s. Consistently.

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This hows the clear future of mathematics in empirical, experimental maths & those of complex systems.

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Let us look for more information on these patterns in the sequence of the numbers from 21,000 to 21,540, that is 6 groups of 90, which is the natural pattern of the primes. In these we will see the prime numbers steadily declining, and the appearances of the prime gaps, specifically, and how this comes about. How this applies to the patterns of the primes, prime gaps, & even Goldbach’s will then be more clear.

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90’s segment # of primes PM3’s Prime Multiples Totals, always 36

21000 to 11 12 13 36

21090 to 10 12 14

21180 to 8 ” 16

21270 to 8 ” 20

21360 to 8 ” 16

21450 to 10 ” 14

21540 to 11 ” 13

21630 to 10 ” 14

21720 to 10 ” 14

21810 to 10 12 14

21900.

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This analysis by 30’s, and the 1, 3, 7, 9, quartets methods, shows the direct proportions between the Number of primes as they decline, and the increasing number of PM’s which create that decline. The primes gaps are thus clearly created by the PM’s, because the PM3’s stay the same.

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Average # of primes per 90 in this segment is 9.6. Exactly. This is an empirical test of the usual method of estimating the number of primes per segment of the number line. and can be applied anywhere in the number line to test the current estimator’s accuracy, as well. Does the estimate match the outcome, empirically and is there a cognitive dissonance? Not quite, in most cases, and that shows the power of Quartets method, and that the 30’s units needs to be incorporated to constrain and improve the estimates, which are based upon 10’s/100’s groups, not the 90’s, which are the natural prime occurrences. The Procrustean Bed of the decimal system is at work, here. & That needs to be corrected to improve the estimator, to some extent.

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Thus the decreasing primes in the number line are due exactly to the increasing PM’s of the 1, 3, 7, 9 number line as well, when compared and measured using the 30’s cycle of the PM3 quartets, which drives the patterns of the primes.

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And we see when we add up the number of 2 primes to create even numbers, those numbers with large 3 and 6, and mostly 30 divisors also have the most two prime sums creating them. If the quartet system was not used, we’d not been aware of why that was, either.

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And that explains the 30’s difference among most of the primes, as well, as noted in the tables above.

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And because -3 tends to be more often followed by -9 ending primes (Soundararajan and Oliver, 2016), the 3’s predilection is very easily explained by the patterns of 3 multiples found in the primes.