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quick observation about primes


moth
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While looking for a modular building block for a sieve of Eratosthenes, for example, an 8-bit block of zeros with composite multiples of 2 and 3 already marked as ones,I noticed something bazaar.
If I grouped the digits from the sieve into groups of 8 and marked the composites, it takes the same number of groups of eight to form a repeatable block as the prime who's composites i was marking.
For example, here are some primes and the groups of eight digits for the sieve.

2  == 10101010
3  == 10010010 01001001 00100100
5  == 10000100 00100001 00001000 01000010 00010000
7  == 10000001 00000010 00000100 00001000 00010000 00100000 01000000
11 == 10000000 00010000 00000010 00000000 01000000 00001000 00000001 00000000 00100000 00000100 00000000
13 == 10000000 00000100 00000000 00100000 00000001 00000000 00001000 00000000 01000000 00000010 00000000 00010000 00000000
17 == 10000000 00000000 01000000 00000000 00100000 00000000 00010000 00000000 00001000 00000000 00000100 00000000 00000010 00000000 00000001 00000000 00000000
19 == 10000000 00000000 00010000 00000000 00000010 00000000 00000000 01000000 00000000 00001000 00000000 00000001 00000000 00000000 00100000 00000000 00000100 00000000 00000000

For a little more context:
Start with the rightmost digit representing one.
The digit to the left represents 2 the digit to the left of that represents 3 etc.
the digit representing one is left a zero and multiples of 2 and 3 are marked with a 1.
A string of digits is repeatable if it has a 1 as the leftmost digit and a zero on the right.

To make a repeatable block for 2 and 3:

             2 == 10101010 10101010 10101010
             3 == 10010010 01001001 00100100
building block == 10111010 11101011 10101110

Now the building block can be repeated over and over and only the zeros on the list (except 1 of course) are potential primes.

I thought it was cool that (as far as i can tell) no matter what size blocks you use 8,5 ,64 whatever, the prime self-references in the number of groups like that.

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While looking for a modular building block for a sieve of Eratosthenes, for example, an 8-bit block of zeros with composite multiples of 2 and 3 already marked as ones,I noticed something bazaar.

If I grouped the digits from the sieve into groups of 8 and marked the composites, it takes the same number of groups of eight to form a repeatable block as the prime who's composites i was marking.

For example, here are some primes and the groups of eight digits for the sieve.

2  == 10101010
3  == 10010010 01001001 00100100
5  == 10000100 00100001 00001000 01000010 00010000
7  == 10000001 00000010 00000100 00001000 00010000 00100000 01000000
11 == 10000000 00010000 00000010 00000000 01000000 00001000 00000001 00000000 00100000 00000100 00000000
13 == 10000000 00000100 00000000 00100000 00000001 00000000 00001000 00000000 01000000 00000010 00000000 00010000 00000000
17 == 10000000 00000000 01000000 00000000 00100000 00000000 00010000 00000000 00001000 00000000 00000100 00000000 00000010 00000000 00000001 00000000 00000000
19 == 10000000 00000000 00010000 00000000 00000010 00000000 00000000 01000000 00000000 00001000 00000000 00000001 00000000 00000000 00100000 00000000 00000100 00000000 00000000

For a little more context:

Start with the rightmost digit representing one.

The digit to the left represents 2 the digit to the left of that represents 3 etc.

the digit representing one is left a zero and multiples of 2 and 3 are marked with a 1.

A string of digits is repeatable if it has a 1 as the leftmost digit and a zero on the right.

 

To make a repeatable block for 2 and 3:

 

             2 == 10101010 10101010 10101010
             3 == 10010010 01001001 00100100
building block == 10111010 11101011 10101110

Now the building block can be repeated over and over and only the zeros on the list (except 1 of course) are potential primes.

 

I thought it was cool that (as far as i can tell) no matter what size blocks you use 8,5 ,64 whatever, the prime self-references in the number of groups like that.

 

I am glad you found this. These are what are called Palindromic primes.

 

 

 

A palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the numbering system and its writing conventions, while primality is independent of such concerns.

 

 

In binary, the palindromic primes include the Mersenne primes and the Fermat primes. All binary palindromic primes except binary 11 (decimal 3) have an odd number of digits; those palindromes with an even number of digits are divisible by 3. The sequence of binary palindromic primes begins (in binary):

11, 101, 111, 10001, 11111, 1001001, 1101011, 1111111, 100000001, 100111001, 110111011, … (sequence A117697 in OEIS)

Due to the superstitious significance of the numbers it contains, the palindromic prime 1000000000000066600000000000001 is known as Belphegor's Prime, named afterBelphegor, one of the mythical seven princes of Hell. Belphegor's Prime consists of the number 666, on either side enclosed by thirteen zeroes and a one.

http://en.wikipedia.org/wiki/Palindromic_prime

Edited by Unity+
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I apologize, Unity+, I didn't explain what I'm doing very well, and chose a poor example, but the main point is that these blocks of ones and zeros are not meant to be prime.
They are pieces of a http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes.

The rightmost 'digit' represents 1, each digit to the left of the first represents it's position on the list. 2 is second,3 is third on the list etc..
If I was using the sieve to find all primes less than 10 million, it will take a list 10 million digits long.

The 8 digit group with 2,4,6, and 8 marked with a 1 looks like:10101010 and that pattern repeats for the rest of the digits in the sieve.
Since multiples of a prime are not prime you only have to check the digits marked '0' for primes.

I'm looking at combining many patterns, like the 1010 pattern for 2 with the next primes: 3,5,7 etc. to form a modular block that can be repeated over and over for a sieve with most of the composites already marked.

Belphegor's Prime may be my next pseudonymn.

Edited by moth
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