[INTJ] I may have a new Sieve for Prime Numbers. I think it is kind of fun to do.

I may have a new Sieve for Prime Numbers. I think it is kind of fun to do.

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  • 4 Post By functionaloneness
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This is a discussion on I may have a new Sieve for Prime Numbers. I think it is kind of fun to do. within the INTJ Forum - The Scientists forums, part of the NT's Temperament Forum- The Intellects category; I suggest trying this out on your own, even if you don't think prime numbers are interesting, give it a ...

  1. #1

    I may have a new Sieve for Prime Numbers. I think it is kind of fun to do.

    I suggest trying this out on your own, even if you don't think prime numbers are interesting, give it a shot. You don't even need to know if 101 is s a prime number or not.

    All prime numbers greater than 3 are multiples of 6 + or -1.
    You can do 6's. just write them out with enough space to write some number in between. Like this: 5 6 7 11 12 13 17 18 19 23 24 25 29 30 31 35 36 37 ... just keep going if you can to 100 or 150. but if you can make your rows begin with 5, 35, 65. 95 and so on. Go high enough to make it challenging. The even numbers should all line up so you can cross them out, and the 2 columns of 5, but we'll come back to those.

    This contains all the prime numbers to whatever number that you went to, but of course all of those aren't prime numbers. Let's weed those out. 5's are easy to eliminate but let's use another way that will show the process of this sieve.
    Take 7 and multiply it times 6 which gives you 42. Starting with the prime number 7, keep adding 42 till you reach the amount you went to with your 6's. Like this 7 49 91 133 175 217 ... You can cross these off of your list

    Do the process with 11. 11 x 6 is 66. So add 66 to 11 and each following number. 11 77 143 209 ...

    Now for 13. 13 x 6 + 78. So add 78 to 13 and keep adding.

    On paper, when you write these out and use those numbers in your first column for each row of numbers. That column other than the first number, the prime number, is a column of elimination. It matches with the composite (non- prime) numbers on the list.

    But there is more. As you noticed with 5, there are two columns of numbers you can eliminate, a 2 for 1 deal. the other number is the square of the prime number. That happens for every prime number that falls in the sequence of 5 11 17 23 29 35 41 47 53 and so on

    What is also cool is that rather than adding all those numbers, 6 times the prime number to the prime number, they are actually multiples of the prime number: 1 7 13 19 25 31 37 43 49 55 61.

    Do those 2 sequences look familiar? Put them together and you have 5 7 11 13 17 19 23 25 29 31 35 37 41 43 47 49.

    All of the numbers identify the numbers on the list that don't belong, and show a fascinating order to prime numbers.

    I have done this up to 2000 and it has not missed identifying each composite number on the list.
    I hope you enjoyed this.



  2. #2
    ENFP - The Inspirers

    I love that someone is even trying to sieve primes. This is why I like INTJs. I'll try it out (and maybe my kid will enjoy it too - he is a Mathy ENTP) and come back to you on this.

    But really happy to see such things on this forum! Lovely stuff. 🙂

  3. #3

    Nice. I may be wrong but i thought there are already many advanced sieves algorithm to calculate prime numbers. And other algorithm to find the largest prime number, GIMPS one of them.

    Maybe you can elaborate on the efficiency level, have you calculate it?


    ewdenore and functionaloneness thanked this post.

  4. #4

    Quote Originally Posted by contradictionary View Post
    Nice. I may be wrong but i thought there are already many advanced sieves algorithm to calculate prime numbers. And other algorithm to find the largest prime number, GIMPS one of them.

    Maybe you can elaborate on the efficiency level, have you calculate it?


    I believe that most Sieves deal with all numbers and multiples. I don't know of any that deal with just numbers derived from multiples of 6 + or -1. This cuts the numbers by 2/3. You are only dealing with all the numbers that could be prime numbers. If you make the table for the Prime Number 5 (the first number on each row would be 5 35 65 95...) using all the numbers, what you get is quite fascinating.

    The process I use identifies each composite number on the this list, no extra numbers, no numbers missed. It also very simple and easy to use. Each Prime number creates 1 or 2 columns (or calculations) that uniquely identify each composite number on that list.
    Let's identify all the composite numbers up to 200 on this list (5 7 11 13 17 19 23 25 29 31 35 37 41 43 47 49 53 55 59 61...)

    Start with 5. 5 x 6 = 30. So we will add 30 to 5 and each row or following sum.
    35 65 95 125 155 185

    Since the square of 5 (25) does't fall into this set of numbers it created another row or list of composite numbers. We continue to add 30
    25 55 85 115 145 175

    For 7, add 42
    49 91 133 (175)
    I consider the multiples of 5 act as spacers for either where a row starts or to continue the sequence of composite numbers

    For 11, add 66
    77 143

    For the square of 11
    121 187

    For 13, add 78 starting with the square of 13 (169)
    169

    For 17, add 102
    119
    The square of 17 would be included but it is greater than 200

    For 19, its square is also greater than 200

    For 23, add 138
    161

    There are no more composite numbers on the list to 200.
    Results: There are 22 composite number on our list to 200.
    This process identified all 22.

    175 was repeated but that was the result of multiplying the prime number by a multiple of 5. By adding we are just multiplying the prime number by this sequence.
    7 13 19 25 31 37 43 49 55 61

    We have 2 columns (lists) of composite numbers with this sequence:
    5 11 17 23 29 35 41 47 53 59

    It is a little mind blowing that the problem and the solution come from the exact set of numbers.
    Last edited by functionaloneness; 12-16-2018 at 10:23 PM.
    ewdenore and contradictionary thanked this post.


 

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