June 19th, 2017, 05:25 PM  #11 
Senior Member Joined: May 2016 From: USA Posts: 1,084 Thanks: 446 
"The string" strongly implies one string. "Infinite number of strings" implies more than one. As I said before, you may have something interesting to say (whether novel or not), but there is no way to tell. You seemed to have started with a cognizable proposition that was interesting, but completely wrong. What exactly are you now proposing for general contemplation, discussion, etc? As far as I can tell, what you are saying is that if a composite number does not have 3 as a factor, it has a larger prime as a factor. I agree wholeheartedly. 
June 21st, 2017, 12:46 PM  #12 
Newbie Joined: Apr 2016 From: Arizona Posts: 19 Thanks: 0 
When I saw this pattern, that shifts when every P^2 entered the equation, I started looking for a graph that would show that the prime aren't random, but that they are controlled by the composites as the pattern changes. Like everyone else I started by leaving in the "interference" of the even numbers. No go. Just pandemonium. I then went to a graph using just the odd numbers from 1 to 49 on the top row. on the second row 51 to 99. I took my graph to seven rows. Then I marked all the "3" composites on the first row and the first one on the second row (51). Starting at the number 3, I dropped to the next row (53), and moved backward to 51. this showed me the pattern for the "3" composites. Going to 15, drop to the next row, back one put it at 63. Also, 15, drop to the next row and forward 2 numbers is 69. The "7" composites worked just as well. 7+508=49. 7+50+6=63. In other words, down one row, back four numbers, or forward three, and the "7" composites are found. There was a pattern for each of the prime/composites on my graph up to 7*49. If the top row is expanded to 1to 99, or higher, the pattern shifts, but it is easy to find. I thought it was interesting, although useless, information, as it isn't useable to detect the very high composites and primes.

June 21st, 2017, 04:37 PM  #13  
Senior Member Joined: Aug 2012 Posts: 1,973 Thanks: 551  Quote:
It's entirely true that the primes are not random in the sense of being undetermined. Martians have the same prime numbers we do. The primes are in some way logically necessary, once we accept the counting numbers and define what we mean by a prime. The distribution of the primes may or may not be statistically random. A lot of people try to understand this problem. There are definitely patterns in the way the primes are sieved. You throw out all the multiples of 2, then of 3, then of 5, and so forth, and of course there must be some clever way to express the resulting pattern. People have been driven by this thought for centuries. A formula that gives the nth prime without having to calculate all the earlier ones. Investigating this subject is a worthy endeavor. I think if you challenged yourself to describe your idea more clearly, people could give more specific constructive advice. Last edited by Maschke; June 21st, 2017 at 04:53 PM.  
June 23rd, 2017, 01:02 PM  #14 
Newbie Joined: Apr 2016 From: Arizona Posts: 19 Thanks: 0 
I only discard the even numbers from my graph. The top row consists of all the odd numbers from 1 through 49. Once the first composite sequence for the prime to composite is found, it holds true for all the other composites of that prime. Ex. 3+48=51, 15+48=63, 21+48=69, and so on up the line. 7+42=49, 21+42=63, etc. Each prime after 5 can be calculated before it reaches P^2, at which point it changes the composite pattern, thus changing the prime pattern. If the top row of the graph is changed to 1through 99, or higher, the pattern for each prime/ composite will shift. 3+102=105, so adding 102 to all the 3 composites will run true on that graph. 7+98= 105. adding 98 to every 7 composite will also run true on that graph. As I said, interesting but useless in the search for the high end primes.

October 27th, 2017, 05:22 PM  #15 
Newbie Joined: Apr 2016 From: Arizona Posts: 19 Thanks: 0 
The only way to understand what I am trying to say, and to determine if my statements are true or false, is to try the "revised sieve" yourself. There are two facts I will state; Every line (or number string) formed by a prime(or composite) will start at 3*P or C. 3*1P, 3*3C, 3*5C, 3*7C, and every 3*odd number, in sequence from there. 5*1P, 5*3C, *5C, 5*7C, and again, every odd number in sequence from there. Take any odd number(prime or composite) and you will get a number string with the same sequence; 1*P or C, 3*P or C, 5*P or C, and up through the odd numbers in sequence from there. On a "sieve" with the Top line of 50 (odd numbers 1 through 49 inclusive),and a line of 50 below it (51 through it), you should see for 3P, and every 3C on the top line add 50 (that will make 53 on the next line down). From 53 move 1 number back to51(a composite of 3). This is the pattern of the composites formed by 3P. If you use a top line of 100 (1 through 99, inclusive) the pattern for the composites formed by three will move forward 1 number from 3, or composite of three, (3+100= 103 plus 1 odd number forward) to 105. For 5 and its composites the pattern will be straight down the columns if you use a top line in multiples of 50 (50, 100, 150, etc.). The composites formed by any prime or composite will form a pattern that remains the same up through the "sieve" as far as you want to take it. I don't ask anyone to believe what I have posted. There is a lot of false info on the internet and in books. The only way to determine if I am telling the truth, or not, is to try this for yourself. If you try it , and determine I am wrong, you will know for yourself it is false. If, on the other hand, you determine I am right, maybe you, with your higher mat skills (I only went to geometry and didn't understand much of that or algebra) could make a useful equation out of it. With that lengthy explanation, I will leave this site' as my meager math skills are of no use here. 'Bye to everyone and have a nice day!

January 21st, 2018, 12:19 PM  #16 
Newbie Joined: Apr 2016 From: Arizona Posts: 19 Thanks: 0 
The first prime pattern starts as 1,2,3. I am not calling 1 a prime, but it is the first number. At 2^2, all the even numbers are composites, so only the odd numbers have a chance to be prime. This sets the second pattern. Moving up 3 odd numbers is the next composite, 3^2=9. Thus, 5 and 7 are prime. 3*5=15 is the next composite, hence 11 and 13 are composite. Now that 5 has entered the list, the pattern will shift when 5^2 is reached. 17 and 19 are prime. the third number after 15 is a composite,3*7=21. but now the pattern shifts. composites of 5 jump into the picture. 5 odd numbers from 15 is 25. that makes23 prime. the dance of 3 and 5 continues until 7^2. 3 odd numbers from 21 is 27. No primes between 25 and 27. 3 odd numbers past 27 is 33. 29 and 31 are prime. 5 odd numbers from 25 is 35. no primes between 33 and 35. 37 is prime, 39 is composite. 41 and 43 are primes, and 3 and 5 meet at 45, composite. now 7 jumps on the stage. 7 odd numbers from 35 is 7^2, or 49. 47 is prime. The pattern has altered again, and will continue this pattern of 3,5,7 dancing around each other until 11^2,ro 121. 9 is a composite of 3, so 9^2 (81 composite) doesn't make the pattern change.

January 21st, 2018, 06:32 PM  #17 
Senior Member Joined: May 2016 From: USA Posts: 1,084 Thanks: 446 
I think you are saying that if $c_i$ is the ith odd composite number and $p_j$ is the jth odd prime, then $c_i \le p_j^2 \implies \exists \ k \in \mathbb N^+ \text { such that } k \le j \text { and } p_k  c_i.$ Is that it? For example, if an odd composite is less than 25, then 3 is a factor. Well, the odd composites < 25 are 9, 15, and 21, all of which are divisible by 3. That seems correct, and a proof seems well within reach, but what does it tell us that the Sieve of Erastothenes had not told us over two millennia ago. 
January 22nd, 2018, 11:10 AM  #18 
Newbie Joined: Apr 2016 From: Arizona Posts: 19 Thanks: 0 
With the Eratosthenes Sieve, the even numbers "cloud" the perspective of the pattern. Using only the odd numbers, the pattern is a little more evident. From 21 and 25 there is 27, 33, 35. the 3 composites move closer to the five composites by one odd number. With two more 3 composites, 39 and 45, the 3 composites meet the 5 composites. 7^2 jumps in at 49, but doesn't upset the 3,5 dance until 63, where 3 and 7 meet. The composite numbers 51,53,57 mimic the pattern of 21,25,27. 63,65,69 copy the pattern of 33,35,39 only because 3 and 7 meet. 75, where 3 and 5 meet again (every 15 odd numbers) and 77 show the 7 composites running away. 81,85,87 show only the 3,5 pattern. 91,93,95,99 shows the 7 composite jump into the 3,5 pattern again. At 105, 3,5,7 meet to form a composite. Then it starts over.3 odd numbers (ON) from 105 is 111. 5 ON from 105 is 115. 6 ON from 105is 117. 7 ON from 105 is 119. 11^2 enters at 121, but is only relevant every 22 whole numbers, or 11 ON. (121+22=143)(123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143. 11ON from 121 to 143)

January 22nd, 2018, 12:47 PM  #19 
Senior Member Joined: May 2016 From: USA Posts: 1,084 Thanks: 446 
Is it possible to express this pattern in mathematical notation? When you say that "with 39 and 45, the 3 composites meet the 5 composites," why is 39 a 5 composite? In what way does 21, 25, 27 mimic 51, 53, 57? Of course I think you mean 51, 55, 57 because 53 is a prime. If n = 3k then n + 30 = 3k + 30 = 3(k + 10). If n = 5k, then n + 30 = 5k + 30 = 5(k + 6). Those are true facts about composites for which 3 or 5 are factors, but what does it say generally about primes? I continue to think that all you are saying is that if p is a prime and x is a composite number less than p squared, then a prime less than p is a factor of x. 
January 24th, 2018, 02:32 AM  #20 
Newbie Joined: Apr 2016 From: Arizona Posts: 19 Thanks: 0 
Sory,JeffM1. I meant to type 51, 55, 57,as I know 53 is a prime. What I am trying to say is that as the composites enter at P^2, the pattern of the primes is changed. Using an odd numbered sieve, instead of a sieve with odd and even numbers, it is easier to find the composites, thereby finding the primes easier and showing the prevailing pattern of the primes at that point. Instead of dividing every odd number by all the primes up to the square of that number and then doing the same process again to prove primality, find the pattern of the composites on the odd number sieve, follow the pattern of each composite made by all the different primes (3^2 up to P^2) for the chart, and then divide all the numbers not shown as a composite by the primes 3 to P to prove primality. Unfortunately, I am over two millennia late, as this isn't advantageous with the high numbers being searched. This idea is biased for the fact that every time the top row in the sieve I described in my earlier post changes the way the pattern of the composites appears. This does not affect the pattern of the primes.


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