February 5th, 2018, 12:08 PM  #21 
Newbie Joined: Apr 2016 From: Arizona Posts: 19 Thanks: 0 
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.

April 13th, 2018, 03:28 AM  #22 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
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.

April 18th, 2018, 03:30 PM  #23 
Newbie Joined: Apr 2016 From: Arizona Posts: 19 Thanks: 0 
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.

April 20th, 2018, 11:16 AM  #24 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
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"!

July 7th, 2018, 02:24 PM  #25 
Newbie Joined: Apr 2016 From: Arizona Posts: 19 Thanks: 0 
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).

July 17th, 2018, 05:31 AM  #26 
Member Joined: Aug 2015 From: Montenegro (Podgorica) Posts: 37 Thanks: 4 
Friend, if I understood you correctly here is the answer, if you add 3 to itself multiple times you get: 3 = 3 = 1*3 3 + 3 = 6 = 2*3 3 + 3 + 3 = 9 = 3*3 = 3^2 3 + 3 + 3 + 3 = 12 = 3*4 3 + 3 + 3 + 3 + 3 = 15 = 3*5 3 + 3 + 3 + 3 + 3 + 3 = 18 = 3*6 3 + 3 + 3 + 3 + 3 + 3 + 3= 21 = 3*7 ... you get 3*n where n goes from 1 through every number including primes; similarly, 7 will multiply every natural number including the odd primes (3, 5) until it reaches 7^2 = 49: 7 = 7 = 1*7 7 + 7 = 14 = 2*7 7 + 7 + 7 = 21 = 3*7 7 + 7 + 7 + 7 = 28 = 4*7 7 + 7 + 7 + 7 + 7 = 35 = 5*7 7 + 7 + 7 + 7 + 7 + 7= 42 = 6*7 7 + 7 + 7 + 7 + 7 + 7 + 7= 49 = 7*7 ... 11 also, until it reaches 11^2 = 121, it will at some point multiply 3 (which is 33), 5 (which is 55), 7 (which is 77) and eventually 11 itself (which is 121), and so on with all the other prime numbers. So I don't see what is the problem with, as you said, "change in pattern". Last edited by skipjack; July 17th, 2018 at 11:26 AM. 
July 19th, 2018, 04:05 PM  #27 
Newbie Joined: Apr 2016 From: Arizona Posts: 19 Thanks: 0 
As most everyone knows, any even number higher than 2 is a composite, o why leave them in and muddy the "water"? But the 5 and 9 composites need to remain on the sieve I described to negate any mistake in counting on the sieve. start with the string of composites 3. At that point, the pattern is every third odd number is a composite. The 5 composites start at 3*5 (15), but it shows as a composite of 3, so there is no change in the pattern at this time. 3 odd numbers further is 3*7 (21). 7 now enters the mix, but still no change in the 3 pattern. 5*5 (25) changes the pattern as it is 2 odd numbers past the last 3 composite. Now run the 3 and 5 pattern up until 3*15 (45) is reached and you will see that the 3,5 pattern repeats every 15 odd numbers. Up to this point, 7 has not formed any composites that change it. 7*7 (49) is the first 7composite to interrupt the pattern. 7odd numbers past 5*7(35) is 49. The 3,5 pattern shows in the 51c, 53p, 55c, 57c, 59 p. Look back at 21c, 23p, 25c, 27c, 29p. Look familiar? That is because 7 doesn't form another composite until 7*9 (63, a composite of 3) No new composite, so 61p, 63c,65c, 67p, 69c look the same as 31p, 33c, 35c, 37p 39c. 11 forms a new composite, but it will first be detected as a 7 composite, so p*p adds a new composite. p*c doesn't add a new composite since that number would be detected sooner in the search. 7 and 11 skip the 80"s so the 80's look peculiar like the 50's. 7 and 11 enter in the 90's (91 and 99). Every time a prime number reaching p*p will not form a new composite until p odd numbers forward from p*p. If the next p odd numbers forward results in p*p, then a new composite forms. If the next p odd numbers forward results in p*c, no new composite formed. To learn the pattern of the primes, the pattern of the composites must be found, as with the original sieve.

July 20th, 2018, 05:54 AM  #28  
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551  Quote:
In the ordered set of numbers that 3 divides evenly, it is true that every third odd number is composite, but of course every number in that set except the first is composite. You may have something interesting thing to say, but until you take the care to say it exactly, it is just drivel.  
July 20th, 2018, 06:45 AM  #29 
Newbie Joined: Sep 2017 From: Belgium Posts: 19 Thanks: 7 
I think you just noticed some effect of the cyclicity of primorals $p_a$# (when number are sieved up to $p_a$, see also thread on "prime number sequence") and the fact that we look for primes only up to $\sqrt x$ when sieving. You can find a lot of patterns in primes (they are not random at all), especially when looking at primorals (6n+x, 30n+x, 210n+x,...) 

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