# Some Properties and Algorithms for Twin Primes
## 4 Another Algorithm for Twin Primes  
In Appendix A Section [1], we start by writing the following Algorithm 2 for twin primes: \begin{tabular}{c l} \hline \hline
**Algorithm 2:** Algorithm for twin primes \\ \hline (1) & Start with a list of initial known primes, \(P=\) [2,3,5], if the list is empty \\ (2) & Define the initial generator value \(G\) and the generator index \(I\). \\ (3) & Compute the next generator value \(G_{next}\) using the current generator \(G\) and the next prime in the list \(P\) \\ \(G_{next}=G\times G_{I+1}\) \\ (4) & Construct a set \(R\) of primes such that \(p\in P\) and \(p^{2}<G_{next}\) \\ (5) & Generate a new list of potential primes: \\ & - Generate a set \(F\) of potential new primes using the following families: \\ & - From existing primes: \\ & - For each \(p\in P\) and \(p\
eq P_{i}\), calculate potential primes: \\ & \(\text{For }j=1,2\ldots\text{: }F=(G\times j-p)if(G\times j-p<G_{next})\) \\ & - From previously removed numbers: \\ & - For each \(r\in R\), calculate potential primes: \\ & \(\text{For }j=1,2\ldots\text{: }F=(G\times j-r)if(G\times j-r<G_{next})\) \\ & - With adjustments of \(-1\) and \(+1\): \\ \(F=(G\times j\mp 1)\) \\ (6) & Optimize the list of potential primes: \\ & - Remove numbers from \(F\) that are multiples of any prime in \(R\) \\ & \(F=F\{f\in F|\exists p\in R:f\bmod p=0\) \\ (7) & Merge the optimized list of potential primes into the main list of known primes \\ (8) & Clean the list of known primes to remove any invalid entries \\ (9) & Repeat the process with updated parameters to continue generating primes. \\ \hline \hline \end{tabular}   
We generate a total of 896.062 pairs of twin prime numbers as in Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline _Index_ & **P1** & **P2** \\ \hline \(0\) & \(3\) & \(5\) \\ \hline \(1\) & \(5\) & \(7\) \\ \hline \(2\) & \(11\) & \(13\) \\ \hline \(3\) & \(17\) & \(19\) \\ \hline \(4\) & \(29\) & \(31\) \\ \hline \(5\) & \(41\) & \(43\) \\ \hline \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \hline \(896\),056 & 223,091,501 & 223,091,503 \\ \hline \(896\),057 & 223,091,549 & 223,091,551 \\ \hline \(896\),058 & 223,091,621 & 223,091,623 \\ \hline \(896\),059 & 223,092,047 & 223,092,049 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Generated twin prime pairs—the table shows a selection of the 896,062 twin prime pairs identified in the experiment, including the smallest (3, 5) and the largest (22,309,2671, 22,309,2673). The dataset highlights the algorithm’s efficiency in identifying twin primes across a wide numeric range. Consequently, the first column is the value k, while the second and the third columns are the first and the second elements of a couple of twin prime numbers. Below in Figure 3, we report the graph that relates the prime numbers generated to the time spent generating them, where the prime generated scale is multiplied to \(1\times 10^{7}\). The function presents a logarithmic behavior, which could be an excellent field of application for a Polynomial Regression in search of a possible Fit. The following work is performed in Python in a Jupyter-lab environment (the details are in Appendix A Section [2]). Following a series of trials, as in Figure 4 we found that the level 15 configuration emerged as the most optimal and, in Appendix A Section [3], we proceed to generate the actual model by exploiting the degree just identified. Figure 4: Polynomial regression of degree 15 on prime number generation—the graph illustrates the results of applying a polynomial regression of degree 15 to model the relationship between the time elapsed (in seconds) and the number of primes generated. Figure 3: Relationship time—prime numbers generated and the number of prime numbers generated. The curve illustrates how the number of primes generated increases over time, eventually reaching a plateau as time progresses.