Just as a prime has a twin prime, there is a "consecutive twin prime" in a twin prime. For example, a set of twin primes whose difference is $4$ so that the next twin prime of the twin prime { $5,7$} is { $11,13$} is defined here as "consecutive twin primes". This time, too, prime numbers greater than $5$ are considered.

The following is a program that searches for consecutive twin primes. Twin prime numbers can be determined by the following four equations.

$$\begin{eqnarray} n&=&6 s_1^2+(6 s_1-1)(m_1-1)\tag{1a} \\ n&=&6 s_2^2+6 s_2+1+(6 s_2+1)(m_2-1)\tag{1b} \\ n&=&6 s_3^2-2 s_3+(6 s_3-1)(m_3-1)\tag{1c}\\ n&=&6 s_4^2+2 s_4+(6 s_4+1)(m_4-1)\tag{1d} \end{eqnarray}$$

$$\begin{equation}n=1,2,3\cdots\end{equation}$$

Give these four equations the same integer $n$. When there is no integer solution for $s$ and $m$, a twin prime is born from the integer $n$. In equation ( $1$), $n$ is expressed as $n_t$ when there is no integer solution for $s$ and $m$. Using $n_t$, the twin prime is given by equation ( $2$).

$$\begin{equation} {\rm Twin\ prime}= \left \{ \begin{array}{l} 6n_t-1 \\ 6n_t+1 \end{array} \right.\tag{2} \end{equation} $$

$$\begin{equation}n_t: {\rm In\ Equation\ (1),{\it n}\ when\ there\ is\ no\ integer\ solution\ for\ {\it s}\ and\ {\it m}.}\end{equation}$$

Now consider ( $n_t + 1$). If $n_t$ is a twin prime number and ( $n_t + 1$) is also a twin prime number, the difference is $(6 (n_t + 1) -1) -(6 n_t + 1) = 4$.

Here is a program that searches for consecutive twin primes. The frequency of consecutive twin primes is quite low, so we expanded the search range from $n$ to $(n + 1000)$. As a result, calculation time may be required. You may not be able to search at once. In that case, enter $n$ again according to the instructions.

n = 1;

Clear[nt1, nt2, nd];
If[n > 0 && IntegerQ[n],
  Do[{dat11 = 
     FindInstance[
      6*s11^2 + (6*s11 - 1)*(m11 - 1) == n && 0 < s11 && 
       0 < m11, {s11, m11}, Integers];
    dat12 = 
     FindInstance[
      6*s12^2 + 6*s12 + 1 + (6*s12 + 1)*(m12 - 1) == n && 0 < s12 && 
       0 < m12, {s12, m12}, Integers];
    dat13 = 
     FindInstance[
      6*s13^2 - 2*s13 + (6*s13 - 1)*(m13 - 1) == n && 0 < s13 && 
       0 < m13, {s13, m13}, Integers];
    dat14 = 
     FindInstance[
      6*s14^2 + 2*s14 + (6*s14 + 1)*(m14 - 1) == n && 0 < s14 && 
       0 < m14, {s14, m14}, Integers]};

   nn = n + 1;

   {dat21 = 
     FindInstance[
      6*s21^2 + (6*s21 - 1)*(m21 - 1) == nn && 0 < s21 && 
       0 < m21, {s21, m21}, Integers];
    dat22 = 
     FindInstance[
      6*s22^2 + 6*s22 + 1 + (6*s22 + 1)*(m22 - 1) == nn && 0 < s22 && 
       0 < m22, {s22, m22}, Integers];
    dat23 = 
     FindInstance[
      6*s23^2 - 2*s23 + (6*s23 - 1)*(m23 - 1) == nn && 0 < s23 && 
       0 < m23, {s23, m23}, Integers];
    dat24 = 
     FindInstance[
      6*s24^2 + 2*s24 + (6*s24 + 1)*(m24 - 1) == nn && 0 < s24 && 
       0 < m24, {s24, m24}, Integers]};

   If[dat11 == {} && dat12 == {} && dat13 == {} && dat14 == {}, 
    nt1 = n, nt1 = "There is no solution in this section."];

   If[dat21 == {} && dat22 == {} && dat23 == {} && dat24 == {}, 
    nt2 = nn, nt2 = "There is no solution in this section."];

   If[dat11 == {} && dat12 == {} && dat13 == {} && dat14 == {} && 
     nt1 > 0 && dat21 == {} && dat22 == {} && dat23 == {} && 
     dat24 == {} && nt2 > 0, Break[]], {n, n, n + 1000}];

  Print["====== The calculation results are shown below. ======"];
  Print["n= ", n];
  Print["nt(1)= ", nt1];
  Print["nt(2)= ", nt2];
  If[IntegerQ[nt1] && IntegerQ[nt2], nd = nt2 - nt1, nd = 0];

  If[nd > 0, Print["Twin prime(1)= ", {6*nt1 - 1, 6*nt1 + 1}], 
   Print["Twin prime(1)= There are no twin primes in the interval,at \
(n)~(n+1000)."]];

  If[nd > 0, Print["Twin prime(2)= ", {6*nt2 - 1, 6*nt2 + 1}], 
   Print["Twin prime(2)= There are no twin primes in the interval,at \
(n)~(n+1000)."]];

  If[nd > 0, 
   Print["PrimeQ= ", {PrimeQ[6*nt1 - 1], PrimeQ[6*nt1 + 1], 
     PrimeQ[6*nt2 - 1], PrimeQ[6*nt2 + 1]}], 
   Print["PrimeQ= {-,-,-,-}"]];

  If[nd > 0, 
   Print["If you want to find the next consecutive twin primes, start \
with n= ", nt1 + 1], 
   Print["If you want to continue, start with n= ", n + 1001]],
  Print["***** ! Please enter a positive integer. *****"]];

This is an example of $n = 1$, but the calculation result is as follows.

====== The calculation results are shown below. ======
n= 1
nt(1)= 1
nt(2)= 2
Twin prime(1)= {5,7}
Twin prime(2)= {11,13}
PrimeQ= {True,True,True,True}
If you want to find the next consecutive twin primes, start with n= 2

$n$ is the input value, and $nt (1)$ is $n$ that satisfies the condition of being a twin prime. $nt (2)$ is $n$ that satisfies the condition that $1$ is added to $nt (1)$ and it is a twin prime. The first twin prime is shown as ${\rm Twin \;prime}\; (1) =$ { $5,7$}. The second twin prime is shown as ${\rm Twin \;prime}\; (2) =$ { $11,13$}. The difference is $11-7 = 4$, which is a continuous twin prime number. "${\rm PrimeQ} $" has confirmed that the four numbers are prime. If you continue the calculation, it indicates $n$ to start next.

Let's check with a slightly larger number. Try $n = 3141592654$. This is not a very large number, but consecutive twin primes will not hit easily. I recalculated $11$ times and finally found it.

====== The calculation results are shown below. ======
n= 3141602664
nt(1)= 3141603117
nt(2)= 3141603118
Twin prime(1)= {18849618701,18849618703}
Twin prime(2)= {18849618707,18849618709}
PrimeQ= {True,True,True,True}
If you want to find the next consecutive twin primes, start with n= 3141603118

The answers were { $18849618701,18849618703$} and { $18849618707,18849618709$}. It started from $n = 3141592654$, but it seems that there were over $10,000$ Barren zones.

For reference, the following table shows the continuous twin prime numbers included in the interval from $n = 1$ to $n = 101217$. One thing you will notice immediately after looking at this table is that the last digit of $n$ is limited to $"2"$ or $"7"$, except when $n = 1$. This is a matter of course if you think a little. By applying this, I was able to write a program that could calculate a little faster, but I was too late to notice.

\begin{array}{|c|c|c|} \hline n\;(=nt\;(1))& {\rm Twin \ prime\ (1)}&{\rm Twin \ prime \ (2)}\\ \hline 1 & \text{{5,7}} & \text{{11,13}} \\ 2 & \text{{11,13}} & \text{{17,19}}\\ 17 & \text{{101,103}} & \text{{107,109}}\\ 32 & \text{{191,193}} & \text{{197,199}}\\ 137 & \text{{821,823}} & \text{{827,829}}\\ 247 & \text{{1481,1483}} & \text{{1487,1489}} \\ 312 & \text{{1871,1873}} & \text{{1877,1879}} \\ 347 & \text{{2081,2083}} & \text{{2087,2089}}\\ 542 & \text{{3251,3253} }& \text{{3257,3259}} \\ 577 & \text{{3461,3463}} & \text{{3467,3469}}\\ 942 & \text{{5651,5653}} & \text{{5657,5659}}\\ 1572 & \text{{9431,9433}} & \text{{9437,9439}}\\ 2167 & \text{{13001,13003}} & \text{{13007,13009}}\\ 2607 & \text{{15641,15643}} & \text{{15647,15649}}\\ 2622 & \text{{15731,15733}} & \text{{15737,15739}}\\ 2677 & \text{{16061,16063}} & \text{{16067,16069}}\\ 3007 & \text{{18041,18043}} & \text{{18047,18049}}\\ 3152 & \text{{18911,18913}} & \text{{18917,18919}}\\ 3237 & \text{{19421,19423}} & \text{{19427,19429}}\\ 3502 & \text{{21011,21013}} & \text{{21017,21019}}\\ 3712 & \text{{22271,22273}} & \text{{22277,22279}}\\ 4217 & \text{{25301,25303}} & \text{{25307,25309}}\\ 5287 & \text{{31721,31723}} & \text{{31727,31729}}\\ 5807 & \text{{34841,34843}} & \text{{34847,34849}}\\ 7297 & \text{{43781,43783}} & \text{{43787,43789}}\\ 8557 & \text{{51341,51343}} & \text{{51347,51349}}\\ 9222 & \text{{55331,55333}} & \text{{55337,55339}}\\ 10497& \text{{62981,62983}} & \text{{62987,62989}}\\ 11202& \text{{67211,67213}} & \text{{67217,67219}}\\ 11582 & \text{{69491,69493}} & \text{{69497,69499}}\\ 12037 & \text{{72221,72223}} & \text{{72227,72229}}\\ 12877 & \text{{77261,77263}} & \text{{77267,77269}}\\ 13282 & \text{{79691,79693}} & \text{{79697,79699}}\\ 13507 & \text{{81041,81043}} & \text{{81047,81049}}\\ 13787 & \text{{82721,82723}} & \text{{82727,82729}}\\ 14802 & \text{{88811,88813}} & \text{{88817,88819}}\\ 16307 & \text{{97841,97843}} & \text{{97847,97849}}\\ 16522 & \text{{99131,99133}} & \text{{99137,99139}}\\ 16852 & \text{{101111,101113}} & \text{{101117,101119}}\\ 18307 & \text{{109841,109843}} & \text{{109847,109849}}\\ 19422 & \text{{116531,116533}} & \text{{116537,116539}}\\ 19882 & \text{{119291,119293}} & \text{{119297,119299}}\\ 20367 & \text{{122201,122203}} & \text{{122207,122209}}\\ 22577 & \text{{135461,135463}} & \text{{135467,135469}}\\ 24027 & \text{{144161,144163}} & \text{{144167,144169}}\\ 26212 & \text{{157271,157273}} & \text{{157277,157279}}\\ 27617 & \text{{165701,165703}} & \text{{165707,165709}}\\ 27807 & \text{{166841,166843}} & \text{{166847,166849}}\\ 28527 & \text{{171161,171163}} & \text{{171167,171169}}\\ 31272 & \text{{187631,187633}} & \text{{187637,187639}}\\ 32477 & \text{{194861,194863}} & \text{{194867,194869}}\\ 32622 & \text{{195731,195733}} & \text{{195737,195739}}\\ 33582 & \text{{201491,201493}} & \text{{201497,201499}}\\ 33637 & \text{{201821,201823}} & \text{{201827,201829}}\\ 36227 & \text{{217361,217363}} & \text{{217367,217369}}\\ 37557 & \text{{225341,225343}} & \text{{225347,225349}}\\ 40007 & \text{{240041,240043}} & \text{{240047,240049}}\\ 40617 & \text{{243701,243703}} & \text{{243707,243709}}\\ 41267 & \text{{247601,247603}} & \text{{247607,247609}}\\ 41332 & \text{{247991,247993}} & \text{{247997,247999}}\\ 42977 & \text{{257861,257863}} & \text{{257867,257869}}\\ 43402 & \text{{260411,260413}} & \text{{260417,260419}}\\ 44447 & \text{{266681,266683}} & \text{{266687,266689}}\\ 44802 & \text{{268811,268813}} & \text{{268817,268819}}\\ 46007 & \text{{276041,276043}} & \text{{276047,276049}}\\ 47457 & \text{{284741,284743}} & \text{{284747,284749}}\\ 47547 & \text{{285281,285283}} & \text{{285287,285289}}\\ 49052 & \text{{294311,294313}} & \text{{294317,294319}}\\ 49312 & \text{{295871,295873}} & \text{{295877,295879}}\\ 49912 & \text{{299471,299473}} & \text{{299477,299479}}\\ 50082 & \text{{300491,300493}} & \text{{300497,300499}}\\ 50332 & \text{{301991,301993}} & \text{{301997,301999}}\\ 54357 & \text{{326141,326143}} & \text{{326147,326149}}\\ 55737 & \text{{334421,334423}} & \text{{334427,334429}}\\ 56822 & \text{{340931,340933}} & \text{{340937,340939}}\\ 57732 & \text{{346391,346393}} & \text{{346397,346399}}\\ 57997 & \text{{347981,347983}} & \text{{347987,347989}}\\ 59042 & \text{{354251,354253}} & \text{{354257,354259}}\\ 59817 & \text{{358901,358903}} & \text{{358907,358909}}\\ 60202 & \text{{361211,361213}} & \text{{361217,361219}}\\ 62542 & \text{{375251,375253}} & \text{{375257,375259}}\\ 64782 & \text{{388691,388693}} & \text{{388697,388699}}\\ 64927 & \text{{389561,389563}} & \text{{389567,389569}}\\ 65377 & \text{{392261,392263}} & \text{{392267,392269}}\\ 65802 & \text{{394811,394813}} & \text{{394817,394819}}\\ 66257 & \text{{397541,397543}} & \text{{397547,397549}}\\ 66292 & \text{{397751,397753}} & \text{{397757,397759}}\\ 67022 & \text{{402131,402133}} & \text{{402137,402139}}\\ 67127 & \text{{402761,402763}} & \text{{402767,402769}}\\
\text{Continue} & \text{Continue} &\text{Continue}\\ \hline \end{array} \begin{array}{|c|c|c|} \hline n\; (=nt\;(1))& {\rm Twin \ prime\ (1)}&{\rm Twin \ prime \ (2)}\\ \hline \text{Continued} & \text{Continued} & \text{Continued} \\ 68672 & \text{{412031,412033}} & \text{{412037,412039}}\\ 69842 & \text{{419051,419053}} & \text{{419057,419059}}\\ 70142 & \text{{420851,420853}} & \text{{420857,420859}}\\ 71207 & \text{{427241,427243}} & \text{{427247,427249}}\\ 73762 & \text{{442571,442573}} & \text{{442577,442579}}\\ 74057 & \text{{444341,444343}} & \text{{444347,444349}}\\ 75422 & \text{{452531,452533}} & \text{{452537,452539}}\\ 77242 & \text{{463451,463453}} & \text{{463457,463459}}\\ 77527 & \text{{465161,465163}} & \text{{465167,465169}}\\ 77912 & \text{{467471,467473}} & \text{{467477,467479}}\\ 78347 & \text{{470081,470083}} & \text{{470087,470089}}\\ 79502 & \text{{477011,477013}} & \text{{477017,477019}}\\ 81762 & \text{{490571,490573}} & \text{{490577,490579}}\\ 82602 & \text{{495611,495613}} & \text{{495617,495619}}\\ 83372 & \text{{500231,500233}} & \text{{500237,500239}}\\ 85102 & \text{{510611,510613}} & \text{{510617,510619}}\\ 86467 & \text{{518801,518803}} & \text{{518807,518809}}\\ 89407 & \text{{536441,536443}} & \text{{536447,536449}}\\ 89462 & \text{{536771,536773}} & \text{{536777,536779}}\\ 89917 & \text{{539501,539503}} & \text{{539507,539509}}\\ 91527 & \text{{549161,549163}} & \text{{549167,549169}}\\ 93202 & \text{{559211,559213}} & \text{{559217,559219}}\\ 93902 & \text{{563411,563413}} & \text{{563417,563419}}\\ 95007 & \text{{570041,570043}} & \text{{570047,570049}}\\ 95442 & \text{{572651,572653}} & \text{{572657,572659}}\\ 97652 & \text{{585911,585913}} & \text{{585917,585919}}\\ 99137 & \text{{594821,594823}} & \text{{594827,594829}}\\ 99612 & \text{{597671,597673}} & \text{{597677,597679}}\\ 101217 & \text{{607301,607303}} & \text{{607307,607309}}\\
\text{Omitted below} & \text{Omitted below} &\text{Omitted below}\\ \hline \end{array}

<<------------------------------------>>

I have posted five reports so far, including this one. All content is related to prime numbers. The target was limited to prime numbers of $5$ or more. If you look at this alone, you may feel like I'm avoiding $"2"$ and $"3"$. I don't hate $"2"$ and $"3"$. I respect this number, $"2"$ and $"3"$.

I used four equations in these reports.

$$\begin{eqnarray} n&=&6 s_1^2+(6 s_1-1)(m_1-1) \\ n&=&6 s_2^2+6 s_2+1+(6 s_2+1)(m_2-1) \\ n&=&6 s_3^2-2 s_3+(6 s_3-1)(m_3-1)\\ n&=&6 s_4^2+2 s_4+(6 s_4+1)(m_4-1) \end{eqnarray}$$

This equation consists only of $"1"$, $"2"$, $"3"$ and $"\tfrac{1}{2}"$. $"2"$ and $"3"$ are both prime numbers. However, $"2"$ and $"3"$ are not only elements of prime numbers, but they feel like numbers with a special role beyond that. I feel that $"2"$ and $"3"$ make rules to determine prime numbers greater than $5$.

Also, in the process of deriving this formula, I could get a glimpse of a geometrically simple structure. This geometrical structure was found in the part related to the prime number of $(6n + 1)$ type and was not recognized in $(6n-1)$ type. I felt the existence of symmetry and asymmetry. I am not a mathematician, so there are limits to the pursuit of these. (I worked as an optical engineer for 50 years and now I retire.)