All primes can be indexed by $k$, as primes must be in the form of
$6k+1$ or $6k-1$. In this paper, we explore for what $k$ such that
either $6k+1$ or $6k-1$ is not a prime. The results can sieve primes
and especially twin primes.
$k \in S_{l} \Rightarrow 6k-1 \not \in \mathbb{P}$, $k \in S_{r}
\Rightarrow 6k+1 \not \in \mathbb{P},$ where $S_{l} = [-I]_{6I+1} =
[I]_{6I-1} \backslash \min([I]_{6I-1}), I \in \mathbb{N},$ and
$S_{r} = [-I]_{6I-1} \cup [I]_{6I+1} \backslash \min([I]_{6I+1}), I
\in \mathbb{N}.$ That is,
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The data set provided source code in C on how to compute Collatz dynamics by automata in terms of residue classes. It also includes algorithms implemented by C codes that can output residue classes by inputting reduced dynamics. The formular for computing a residue class for a given reduced dynamics is as follows:
Function $Invrs(\cdot)$.
$Invrs: c \rightarrow rs$ takes as input \\
$c=O$ or \\
$c=I^{p_1}O^{q_1}I^{p_2}O^{q_2}...I^{p_n}O^{q_n} \in \{I,O\}^{\geq 2},$ $p_i,q_i\in \mathbb{N}^*, i=1,2,...,n, n \in \mathbb{N}^*$\\
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We study a reverse problem - given a reduced dynamics or partial dynamics, can we compute a residue class who presents that dynamics.
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We design a computer program that can randomly generate extremely large integers and output their original dynamics. The source code is txpo10b.c. The bit length of integers can be defined by Macro (named MAXLEN) in source code. The number of randomly generated integers can be set by inputting argument. The program can output the original dynamics of a starting integer in terms of “-” presenting (3*x+1)/2 and “0” presenting x/2. This data can be used for observing the relation between the count of “-” and the count of “0”.
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Currently, the largest integer being verified for Collatz conjecture is about 2^60 . To verify whether extremely large integers such as 2^{100000}-1 can return 1, we design a new algorithm. This dedicated algorithm can change numerical computation into bit or charter computation, hence, original dynamics for extremely large integer without upper bound can be computed. By this algorithm, we thus design computer program that can output original dynamics for extremely large integers without upper-bound such as 2^{100000}-1, which is the largest integer being verified until now.
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We design a computer program that can output reduced dynamics for odd integers with 4k+3, e.g, [3-99999999]. Outputting (reduced) dynamics for much larger integers are also possible. The source code in C is txpo9.c. There are 5 options in arguments for more flexible output. Those data can reveal the properties of reduced dynamics. The most important are ratio and period.
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The data is reduced Collatz dynamics that is denoted by occurred (3*x+1)/2 represented by "-'' and x/2 represented by "0", during the process from a starting integer to the first integer less than the starting integer. The format is: starting integer, binary representation, the count of (3*x+1)/2, the count of x/2, the reduced dynamics represented by "-'' and "0", the ratio - the count of x/2 over the count of (3*x+1)/2.
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