Example 3:

Prediction of the next twin primes after the twin primes pair (1277, 1279)

The location of 1277 is at $\left(i,k\right)=\left(3,6\right)$ . The twin primes $\lambda \left(i,k\right)$ and ${\lambda }^{\prime }=\lambda \left(i,k\right)+1$ are expressed as $\lambda \left(i,k\right)=r_i+210\left(7-1\right)$ . ${\lambda }^{\prime }={r^{\prime }}_i+210\left(7-1\right)$ and ${r^{\prime }}_i=r_i+2$ .

Therefore, $r_i=29$ , ${r^{\prime }}_i=31$ , and $\lambda \left(i,k\right)=1289$ , ${\lambda }^{\prime }=1291$ .

Example 4:

Predicting the longest same-difference primes less than or equal to b

The same-difference in this example means 210.

We can predict the longest same-difference prime series in PTP⁺(0, 10) from the CTC (basic) by checking all rows of Figure S1(a).

For each row i, we find $L_{2311}\left(i\right),i=1,2,3,\cdots ,48$ , below:

$\begin{array}{l}{L}_{2311}\left(1\right)=\left\{2,5,7,9\right\},L_{2311}\left(2\right)=\left\{10,7\right\},L_{2311}\left(3\right)=\left\{11,3,6\right\},\\ L_{2311}\left(4\right)=\left\{10,11,4\right\},L_{2311}\left(5\right)=\left\{11,6,9\right\},L_{2311}\left(6\right)=\left\{10,5,6\right\},\\ L_{2311}\left(7\right)=\left\{10,3,5,6,8,9\right\},L_{2311}\left(8\right)=\left\{10,2,4,8,9\right\},L_{2311}\left(9\right)=\left\{4\right\},\\ L_{2311}\left(10\right)=\left\{2,8\right\},L_{2311}\left(11\right)=\left\{10,11,8,9\right\},L_{2311}\left(12\right)=\left\{10,3,5,7\right\},\end{array}$

$\begin{array}{l}{L}_{2311}\left(13\right)=\left\{11,4,5,8,9\right\},L_{2311}\left(14\right)=\left\{3,5,6\right\},L_{2311}\left(15\right)=\left\{10,11,4,8\right\},\\ L_{2311}\left(16\right)=\left\{10,11,6,7,8,9\right\},L_{2311}\left(17\right)=\left\{10,11,3,4,5,7\right\},\\ L_{2311}\left(18\right)=\left\{10,2,7\right\},\\ L_{2311}\left(19\right)=\left\{11,4,5,6,7,9\right\},L_{2311}\left(20\right)=\left\{11,2,6,7,9\right\},\\ L_{2311}\left(21\right)=\left\{11,3,6,7\right\},\\ L_{2311}\left(22\right)=\left\{10,11,4,9\right\},L_{2311}\left(23\right)=\left\{5,7,8\right\},L_{2311}\left(24\right)=\left\{3,4,6,8\right\},\end{array}$

$\begin{array}{l}{L}_{2311}\left(25\right)=\left\{11,2,3,5,6,7\right\},L_{2311}\left(26\right)=\left\{2,3,9\right\},L_{2311}\left(27\right)=\left\{1,5,8\right\},\\ L_{2311}\left(28\right)=\left\{11,6,7,9\right\},L_{2311}\left(29\right)=\left\{10,11,2,3,7\right\},L_{2311}\left(30\right)=\left\{4,7,9\right\},\\ L_{2311}\left(31\right)=\left\{3,5,6,9\right\},L_{2311}\left(32\right)=\left\{1,10,7\right\},L_{2311}\left(33\right)=\left\{11,4,5,6,9\right\},\\ L_{2311}\left(34\right)=\left\{10,2,4,7\right\},L_{2311}\left(35\right)=\left\{10,11,6,7,9\right\},\\ L_{2311}\left(36\right)=\left\{11,3,4,5,8,9\right\},\end{array}$

$\begin{array}{l}{L}_{2311}\left(37\right)=\left\{10,2,5\right\},L_{2311}\left(38\right)=\left\{1,10,3,4,6,8,9\right\},L_{2311}\left(39\right)=\left\{4,8,9\right\},\\ L_{2311}\left(40\right)=\left\{11,8,9\right\},L_{2311}\left(41\right)=\left\{10,2,7,8\right\},L_{2311}\left(42\right)=\left\{1,10,4,5\right\},\\ L_{2311}\left(43\right)=\left\{11,3,6,8\right\},L_{2311}\left(44\right)=\left\{2,6\right\},L_{2311}\left(45\right)=\left\{2,5,6,7\right\},\\ L_{2311}\left(46\right)=\left\{11\right\},L_{2311}\left(47\right)=\left\{1,3,7,8\right\},L_{2311}\left(48\right)=\left\{10,4,6,8,9\right\}.\end{array}$