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Sorry for the previous question. This is my first post and I didn't write anything special because I wasn't sure if it was worth showing to others. Also, since I am not good at English, I use machine translation. Therefore, please forgive me if there are some parts that are unsightly.

This time, I am prepared to be embarrassed as I write the contents. 1 and 5 are the standard. Add 6 to 1 and 5. Then 7 and 11 will appear first. Adding 6 again yields 13 and 17. Adding 6 again yields 19 and 23. Adding 6 again yields 25 and 29. Adding 6 again yields 31 and 35. By repeating this process endlessly, so far only prime numbers up to 527 have appeared based on this law.

The first thing to pay attention to in this rule is 25, 35, etc. These can be calculated by multiplying 5×5 and 5×7 by the prime numbers that appear in the table. The next thing you should pay attention to is 7×7 and 5×11、11×11. These appear in the table as 49, 55, and 121.

All numbers other than prime numbers that appear in the table can be predicted by multiplying the prime numbers that have appeared in the table up to that point. There is also an order in which numbers other than prime numbers appear, so 7x5, then 7x7, then 7x11, then 7x13, then 7x19, etc. Prime numbers greater than 7, such as 11 and 13, are also 11×5, 13×5, 11×7, 13×7 It will appear on the table as 11×11, 13×13, etc.

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yellow is for prime numbers. For colors other than yellow, for example, red is a multiple of 5, green is a multiple of 7, etc.

The image is calculated by adding 12 based on 1, 5, 7, and 11. However, this is the same as continuing to add 6 based on 1 and 5 mentioned earlier.

If you look at the image, you will understand that there is a law of appearance for all numbers other than the prime numbers that appear in this table.

I don't think it's necessary to say that 2, 3, and multiples of 2 and 3 are not prime numbers. If the law I described holds true, then by combining it with the law for numbers other than prime numbers that appears in the table earlier, it would be possible to calculate all numbers other than prime numbers. Also, as I mentioned earlier, if a number that shares a rule based on two numbers with different standards is a prime number, then it can be explained to some extent why twin prime numbers exist.

I would like to take this opportunity to explain the following four points.

① If you keep adding 6 to each of 1 and 5 as a standard, only prime numbers and numbers multiplied by prime numbers will appear.

② There is a rule for prime numbers that appear by continuously adding 6 to each of 1 and 5 as a standard, and numbers multiplied by prime numbers.

③ If you remove the numbers that satisfy the rule ② from the numbers that appear according to the rule ①, only the prime numbers will remain.

④ By adding rule ② and multiples of 2 and multiples of 3, numbers other than prime numbers can also be predicted.

I would like to know if the above content has already been published or not. As I mentioned earlier, I don't have enough knowledge to know whether this law has already been discovered or not. I even created a table of the ideas that came to my mind and the things I noticed while looking at the prime numbers in a calculation formula. If this already exists, please laugh at it and dismiss it as nonsense from an ignorant person.

Jean Marie
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maki
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    This is a bit hard to follow. I believe you have just noticed that every prime $≥5$ is either of the form $6k-1$ or the form $6k+1$. Starting with $5,1$ you will get every prime $≥5$ if you keep adding $6$. Indeed, you will get every natural number of the form $6k\pm 1$. – lulu Feb 15 '24 at 21:14
  • thank you for your reply. So it was very embarrassing. In my ignorance, I didn't even know if this was a known fact. That's why I asked. – maki Feb 15 '24 at 21:21
  • If I can confirm that this is something that everyone knows as common sense, then I think it was worth taking this opportunity to ask, even if I was prepared to be embarrassed. – maki Feb 15 '24 at 21:37
  • Yes, this is well known. Indeed: every natural number is of the form $N=6k+r$ for some $r\in {0,1,2,3,4,5}$. Now, if $r\in {0,2,4}$ then $N$ is even, hence not prime (if you exclude $N=2$). If $N=3$, then $N$ is divisible by $3$, hence not prime (if you exclude $N=3$). That just leaves $r=1$ and $r=5$. A little work shows that numbers of the form $6n+5$ can also be written as $6k-1$. – lulu Feb 15 '24 at 21:48
  • But don't apologize for the question. There's no harm in asking things. – lulu Feb 15 '24 at 21:49
  • Thank you for teaching me something very valuable. This is a shame, but there is something missing from the above explanation. ⑤All numbers other than prime numbers that appear in the table are composite numbers or parallel numbers of the prime numbers that have appeared in the table up to that point, and there are rules for the appearance of composite numbers of prime numbers. For example, after 7×5, 7×7 appears. There is a rule that the next one is 7x11, the next one is 7x13, the next one is 7x19, the next one is 7x23, the next one is 7x29. – maki Feb 15 '24 at 22:28
  • What I wanted to say was that I was getting carried away thinking that everything could be calculated using the rules ① to ⑤. And of course everyone knows that much! I am very happy that I was able to reconfirm this fact. I would like to express my gratitude once again. Thank you for your reply. – maki Feb 15 '24 at 22:29
  • I asked ChatGPT about what you taught me, and I was able to understand a little bit. I think it's probably just a little. My heart goes out to you. I'm glad I asked and received an answer. I would like to once again express my gratitude. Thank you. – maki Feb 15 '24 at 22:47

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To summarize the discussion in the comments;

Every prime $≥5$ is of the form $6k\pm 1$, so the process described will certainly hit all the primes other than $2,3$ as it hits every natural number of the form $6k\pm 1$.

To see that every prime $≥5$ is of the form $6k\pm 1$, note that the usual division algorithm with remainder tells us that any natural number $N$ can be written as $6n+r$ for some $r\in \{0,1,2,3,4,5\}$. Now, if $r\in \{0,2,4\}$, then $N$ would be even, hence not prime (excluding $N=2$). And if $r=3$, then $N$ is divisible by $3$, hence not prime (excluding $N=3$). Thus if $N$ is prime and $≥5$ we must have $r\in \{1,5\}$. A little caluclation shows that numbers of the form $6n+5$ can also be written as $6m-1$ (just let $m=n+1$), and we are done.

lulu
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