This paper presents a fast and open source extension based on the NSGA-II code stored in the repository of the Kanpur Genetic Algorithms Laboratory (KanGAL) and the adjustment of the selection operator. It slightly modifies existing well-established genetic algorithms for many-objective optimization called the NSGA-III, the adaptive NSGA-III (A-NSGA-III), and the efficient adaptive NSGA-III, (A$^2$-NSGA-III).

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,

We study a reverse problem - given a reduced dynamics or partial dynamics, can we compute a residue class who presents that dynamics.

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”.