Friday, 18 November 2016

Equality of numbers in intervals.




Equality of numbers in intervals.
The interval (0, N) consisting of a number of simple and compound numbers. (0, N) = q + g
q - the number of prime numbers
g - the number of composite numbers
(=) The equality of numbers in intervals
q + g (=) q / + g / equality of numbers in intervals where g = q / number of composite numbers in the lower range, equal to, the number of primes in the high range.
Compose of a number (P_n) - primes such that all numbers in (g) (q), these numbers are equal in intervals (g = q /).
For example, the beginning of a series of 11 numbers
5 + 6 (=) 6 + 7 (=) 7 + 10 (=) 10 + 19 (=) 19 + 48 (=) 48 + 175 (=) 175 + 858 (=) 858 + 5801 (=) 5801+
11 ,,,,,,,, ,,,,,,,,, 13 17 29 ,,,,,,,,,,,, ,,,,,,,,,,,,,, 67 ,, ,,,,,,,,,,,, 223 ,,,,,,,,,,,, 1033 ,,,,,,,,,,,,, 6659 ,,,,,,,,,
Amend the beginning of the series, we will have a different number, different from the first, of the primes. Start row should start from a simple number. It is not part of an existing series, or will simply repeat a series of numbers. For example, the beginning of 19
8 + 11 (=) 11 + 20 (=) 20 + 51 (=) 71 + 282 (=) 282 + 1549 (=) 1549 + 11454 (=) 11454+
,,,,,,,,,,,,, 19 31 71 ,,,,,,,,,,,,,, ,,,,,,,,,,,, 353 ,,,,,,, ,,,,,,, 1831 ,,,,,,,,,,,,,, 13003 ,,,,,,,,,,,,,,,,,,,
And one more number for the example of a prime number 2
2 + 0 (=) 0 + 3 (=) 3 + 2 (=) 2 + 3 (=) 3 + 2
,,,,,,,,,,, 2 3 5 ,,,,,,,,,,,,,, ,,,,,,,,,, 5 ,,,,,,,,,,, 5
And another number for the example of a prime number 7
4 + 3 (=) 3 + 2 (=) 2 + 3 (=) 3 + 2 (=)
Conclusion. When the number of prime numbers greater than the number of composite numbers in the interval, a number is not growing. So, we did the right thing, the beginning of the first series, with the number 11.

We continue. Another series for example, the prime number 23
9 + 14 (=) 14 + 29 (=) 29 + 80 (=) 80 + 329 (=) 329 + 1878 (=) 1878+
23 ,,,,,,,,,,,,,,, 43 ,,,,,,,,,,, ,,,,,,,,,,, 109 409 ,,,,,,,,, ,,,, 2207 ,,,,,,,,,,,,,
What we have three infinite series consisting only of prime numbers and prime numbers in the ranks are not repeated.
1) And such series - infinitely?
Let is on the first question. Because a lot of questions. For example:
2) How to find a prime number (p_n) by its number (n)? Using the formula of the algorithm Sieve of Eratosthenes.
Supplement to the first question. Why are some prime numbers are the number of initial numbers, while others do not, and what's the difference?
Sergey Sitnikov.

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