Good day! I ask for your help – in investigating the twin prime conjecture, I am investigating a simple sawtooth function for the integer factorization of any natural number n = x ^ 2 + k in the range [x^2, (x+1)^2), where k = [0; 2x] (obvious). This function is:
- take the nearest square of the number n as base: x^2,
- add the number k to it, now for every number n = x^2 + k
- fix the same left xleft and right xright part of the equation n = xleft * xright + k
- further, if:
- k < 0 – subtract the number 1 from xleft – this causes k to increase by decreasing xleft * xright
- k > 0 – add ceil(k/xleft) to xright – this causes k to decrease by increasing xleft * xright
- k = 0, we have found a unique way to factorize the number n = xleft * xright, save the result and continue the factorization or stop if xleft is 1 (stopping calculation without result is founded prime number, like in any sieve)
Example of function decomposition:
Base=[ 25 , 36 )
25 0
26 7
27 5
28 3, 7
29 8
30 1, 5, 7
31 8
32 3, 7
33 5
34 7
35 1
IMHO, it is not difficult to prove the hypothesis that in this way we will find all prime numbers and all variants of double factorization of composite numbers in a given range. However, if you look at the final results of this function on different blocks [x^2, (x+1)^2), you can immediately see a repeating pattern of odd number distribution, which immediately seems to be cyclical in this range within each block (two cycles of consecutive odd numbers with an incremental step of 2).
Hypothesis about the universal pattern (sieve) of factorization and determination of prime numbers: If you take a block of numbers between adjacent squares, you can decompose the composite numbers and determine the prime numbers, knowing only the size of this block and nothing else.
If my hypothesis about the existence of a cyclic pattern for factoring numbers in a block is correct, then this leads to interesting results:
- prime and composite numbers actually are defined in the space between the squares of numbers, rather than in the range from 1 to n
- Sieve of Eratosthenes and other global sieves are not the only way to find prime numbers and factorize them
- prime twin conjecture is provable through the study of this new sieve – if an infinite number of block size options can be fed to the sieve, at which prime twins occur, then the number prime twins depends on the number of squares of natural numbers and the new sieve algorithm.
Perhaps someone has a guess on how to at least determine the algorithm of the pattern (potentially sieve) to then try to test it empirically and then prove that it is correct. The first cycle is extremely primitive, but the second and subsequent cycles contain strange small deviations of the pattern for different blocks of squared numbers, the error accumulates and everything breaks.
