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February 5th, 2018, 12:08 PM   #21
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Starting at the beginning, 1,2,3 is the first prime pattern. 1, not a prime, 2 and 3, twin primes (tp). At 2^2, all the even numbers are composite, so the count goes to every,third number is composite. 3^2=9c. 5,7tp. 3*5=15c. 11,13tp. Now the pattern will change every third and fifth number from 15. 3*7=21c. 17,19tp. 5on from 15 s 5^2=25c. 23p. 3*9+27c. No prime between 25 and 27. 3*11=33c. 29,31tp. 5*7 =35c. no prime between 33 and 35. 3on past 33=39c. 37p. 3on past 39 is 45c. 5on from is 45. 3 and 5 meet for a composite so 41, 43tp. 3 and 5 pattern starts again 51c, 55c. But 7^2 enters at 49c, so only 47p and 53p between 45 and 55. 3on from 51 is 57c. 3on from 57 is 63c.7on past 49 is 63c. So 7 and 3 join. This keeps the 3,5 pattern intact through the 60's. 5on past 55 is 65c. 3 past 63 is 69c. 3 past 69 is 75. 5 past 65 is 75. 7 past 63 is 77, so 7 finally enters and up sets the 3,5 pattern. Compare the 20's and the 50's. 21c,51c,23p,53p,25c,55c,27c,57c,29p,59p. 20's and 50's pattern are the same. Compare the 30's and 60's. 31p,61p,33c,63c,35c,65c,37p,67p,39c,69c. Again, the 30's and60's pattern are the same. The 3,5 composite pattern will run the same and meet every 30on. The only change to their pattern is when other primes enter after p^2 and show up again every pon after p^2. 3,5,7 meet at 105c (*35 or 5*21, or 7*15) and run their pattern up to 315c, where they meet again. In those 105on's, 11,13,17 will show up occasionally to upset the pattern and make the primes look random.
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April 13th, 2018, 03:28 AM   #22
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I don't know what you think you are saying but it is obvious, by the definition of "odd", that "odd composites" don't contain any factors of 2! So any such "string" of prime factors has to "start" with 3 times something simply because 3 is the next prime after 2.
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April 18th, 2018, 03:30 PM   #23
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You, Country Boy, and some of the others here see only a small part of what I am trying to say. If 3 were the only odd prime to form composites, the prime pattern would be prime pair, composite, prime pair, repeating as far as anyone wanted to count. But 5 joins in at 3*5=15 (composite). Now the prime pattern changes to every third and fifth number being composite. 3 numbers forward from 15c, 21c, fifth number forward from 15c,25c, sixth number from 15c,27c. Every prime that reaches p^2 will change the prime pattern a little bit. Take 101p for an example. It shows up every 101 odd numbers. 303, 505, etc,, until it reaches 101^2. At that number, it changes the pattern of the primes, but only 101 numbers later and only when it is 101*p. Any composite formed by a prime ending in 3 will follow the pattern outlined in (). (3),(9),1(5),2(1),2(7),3(3). 1(3),3(9),4(5),5(1),5(7),6(3). There is more, but my chemotherapy has me a little tired.
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April 20th, 2018, 11:16 AM   #24
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Essentially you are asking "has anyone noticed that odd composites do not have a factor of 2?" I thought that was pretty much the definition of "odd"!
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July 7th, 2018, 02:24 PM   #25
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What I am trying to say is that there is a pattern to the primes, although it shifts as each prime reaches its square. Take the pattern of 3 and 5. From 15c(3*5) they go their own ways. 7 has not reached its square but it forms a composite with 3 (21). This doesn't upset the 3, 5 pattern. At 7^2 an extra composite is added (7*7=49). The next time 7 appears (63), it doesn't add an extra composite to the mix. So the 3, 5, 7 pattern of the 60's is the same as the 3, 5 pattern of the 30's, as I mentioned in an earlier post. If a prime combines with another prime, after its P^2, then it forms a new composite and messes up the prevailing pattern (ex. 11*13=143). If a prime combines with a composite there is no new composite and the prevailing pattern remains the same (11*15=165).
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