Apparently, the longest same-difference primes is indicated by $L_{2311}\left(46\right)=\left\{11\right\}$ .

Therefore, α is prime, for $\alpha =199+210k$ , $k=\left\{0,1,2,3,4,\cdots ,9\right\}$ .

We then find the longest same-difference prime series in PTP⁺ [0, 10], or a 10-tuple primes series, to be (199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089), as shown on row 46 of Figure 4.

Similarly, we can find the next longest same-difference prime series in PTP⁺ (0, 10), which is associated with $L_{2311}\left(9\right)=\left\{4\right\}$ . Namely, it happens at row 9, listed to be (881, 1091, 1301, 1511, 1721, 1931, 2141), as shown on row 9 of Figure 4.

Example 5:

How many primes are within the interval [2, 2311]?

From the CTCs, we count to find ${\displaystyle {\sum }_{i=1}^{48}\left|L_b\left(i\right)\right|}=188$ for $b=2311$ . From Expression (3), $\pi \left(2311\right)=4+48\left(10+1\right)-188=344$ .

Example 6:

How many twin-prime pairs are within the interval [11, 2311]?

Referring to Expression (4), let $b=2311$ , from CTC, we compute $T\left(i,i+1\right)$ , for $i\in \left\{1,3,6,9,13,16,22,24,30,33,37,40,43,45,47\right\}$ to reach

$T\left(1,2\right)=\left\{k|k+1\notin L_b\left(1\right),k+1\notin L_b\left(2\right)\right\}=\left\{0,2,3,5,7,10\right\}$

Similarly, we have

$T\left(3,4\right)=\left\{0,1,4,6,7,8\right\}$ , $T\left(6,7\right)=\left\{0,1,3,6\right\}$ , and

#Math_498#, and $T\left(47,48\right)=\left\{0,1,4,10\right\}$ .

Therefore, the number of twin-prime pairs is

$\begin{array}{c}{\pi }^{*}\left(2311\right)=3+{\displaystyle \sum }_{i\in w}\left|T\left(i,i+1\right)\right|\\ =\left|T\left(1,2\right)\right|+\left|T\left(3,4\right)\right|+\left|T\left(6,7\right)\right|+\cdots +\left|T\left(47,48\right)\right|+3\\ =6+6+\cdots +4+3=68.\end{array}$