The interval between the twin prime (1) and the next twin prime (2)

Ideas
1

There is no simple rule between the prime number p_n and the next prime number p_{n+1} that cannot be predicted. Similarly, the interval between the twin prime number Tp_n and the next twin prime number Tp_{n+1} cannot be predicted. Considering two twin primes as a set of four primes \{p_{m1},p_{p1},p_{m2},p_{p2}\}, it can be described as Tp_1=\{p_{m1},p_{p1}\} and Tp_2=\{p_{m2},p_{p2}\}.

Here, it is assumed that there are no other twin prime numbers between Tp_1 and Tp_2. The difference between p_{m2} and p_{p1} is defined as "{\rm interval}". When the first prime number p_{m1} is specified, the other three prime numbers are defined by equation (1).

Tp_1= \{p_{m1},p_{p1}\}?{\rm interval}?Tp_2=\{p_{m2},p_{p2}\}
\begin{eqnarray} {\rm interval}&=&p_{m2}-p_{p1}\\ p_{p1}&=&p_{m1}+2\\ p_{m2}&=&p_{p1}+{\rm interval}\tag{1}\\ p_{p2}&=&p_{m2}+2\\ \end{eqnarray}

Look at the following graph.

3459Twinprimegenerationperiodgraph.gif
3459Twinprimegenerationperiodgraph.gif360×255 6.13 KB

The horizontal axis of this graph is the “{\rm interval}” (=p_{m2}-p_{p1}) of the twin prime Tp_1 and the next twin prime Tp_2. The vertical axis is the frequency of occurrence, and the parameter is approximately 61000. What you can see in the graph is the fact that there is a solution only if the “{\rm interval}” is 4,10,16,22,?. In the case of 6,8,12,14,? there is no solution. I will omit the explanation, but this is because one or both of Tp_2 is not a prime number depending on the condition of equation (1). (the sum of odd and odd numbers is even, multiples of 3 are involved)

Here we introduce a variable r representing “{\rm interval}”. In equation (2), r takes an integer value only when “{\rm interval}” is 4,10,16,22, ?.

\begin{eqnarray} r=\frac{1}{6}\times{\rm interval}+\frac{1}{3}\tag{2} \end{eqnarray}
\begin{equation}r=1,2,3\cdots\end{equation}

The purpose of this report is to give r an “{\rm interval}” and search for a set of twin primes that satisfy that condition. 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{3a} \\ n&=&6 s_2^2+6 s_2+1+(6 s_2+1)(m_2-1)\tag{3b} \\ n&=&6 s_3^2-2 s_3+(6 s_3-1)(m_3-1)\tag{3c}\\ n&=&6 s_4^2+2 s_4+(6 s_4+1)(m_4-1)\tag{3d} \end{eqnarray}
\begin{equation}n=1,2,3\cdots\end{equation}

Give the same integer n to these four equations (3). When there is no integer solution for s and m, a twin prime is born from the integer n.

When there are no integer solutions in s and m, n in equation (3) is written as n_t. Using n_t, twin primes are given by equation (4).

\begin{equation} {\rm Twin\ prime}= \left \{ \begin{array}{l} 6n_t-1 \\ 6n_t+1 \end{array} \right.\tag{4} \end{equation}
\begin{equation}n_t: {\rm In\ Equation\ (3),{\it n}\ when\ there\ is\ no\ integer\ solution\ for\ {\it s}\ and\ {\it m}.}\end{equation}

The following program inputs n and r and searches for a set of twin primes that satisfy the specified “{\rm interval}”. Note that “{\rm interval}”=6r-2 from equation (2).

n =    1;
r = 2;

Clear[nt1, nt2, nt3, pm1, pp1, pm2, pp2, pmr, ppr, intr, int1, int2];
n1 = n;
n2 = n + 1;
nr = n + r;
If[n > 0 && IntegerQ[n] && r > 0 && IntegerQ[r],
  Do[
   Do[func[n1];
    If[func[n1] == {{}, {}, {}, {}}, nt1 = n1, nt1 = 0]
     If[func[n1] == {{}, {}, {}, {}}, Break[]], {n1, n1, n1 + 100}];
   
   Do[func[n2];
    If[func[n2] == {{}, {}, {}, {}}, nt2 = n2, nt2 = 0]
     If[func[n2] == {{}, {}, {}, {}}, Break[]], {n2, n2, n2 + 100}];
   
   Do[func[nr];
    If[func[nr] == {{}, {}, {}, {}}, nt3 = nr, nt3 = 0]
     If[func[nr] == {{}, {}, {}, {}}, Break[]], {nr, nr, nr + 100}];
   
   pm1 = 6*nt1 - 1;
   pp1 = 6*nt1 + 1;
   pm2 = 6*nt2 - 1;
   pp2 = 6*nt2 + 1;
   pmr = 6*nr - 1;
   ppr = 6*nr + 1;
   intr = 6*r - 2;
   int1 = pm2 - pp1;
   int2 = pmr - pp1;
   
   n1 = n;
   n2 = n + 1;
   nr = n + r;
   If[int1 == intr && int2 == intr, Break[]], {n, n, n + 100}];
  
  Print["====== The calculation results are shown below. ======"];
  Print["n= ", n];
  Print["r= ", r];
  Print["Specified interval(6*r-2)= ", intr];
  If[int1 == intr && int2 == intr && nt1 > 0, 
   Print["n selected= ", nt1], Print["n selected= -- "]];
  
  If[int1 == intr && int2 == intr && nt1 > 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+100)."]];
  If[int1 == intr && int2 == intr && nt2 > 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+100)."]];
  
  If[int1 == intr && int2 == intr && nt1 > 0 && nt2 > 0, 
   Print["interval(pm2-pp1)= ", int1], "interval(pm2-pp1)= --"];
  
  If[int1 == intr && int2 == intr && nt1 > 0 && nt2 > 0, 
   Print["PrimeQ[(1),(2)]= ", {{PrimeQ[6*nt1 - 1], 
      PrimeQ[6*nt1 + 1]}, {PrimeQ[6*nt2 - 1], PrimeQ[6*nt2 + 1]}}], 
   Print["PrimeQ[(1),(2)]= {{--,--},{--,--}} "]];
  
  If[int1 == intr && int2 == intr && nt1 > 0 && nt2 > 0, 
   Print["If you want to find the next twin prime, start with n= ", 
    nt1 + 1], 
   Print["If you want to continue, start with n= ", n + 101]], 
  Print["***** ! Please enter a positive integer. *****"]];

"********************************************************";
func[n_] := 
 Module[{dat1, dat2, dat3, dat4}, 
  dat1 = FindInstance[
    6*s1^2 + (6*s1 - 1)*(m1 - 1) == n && 0 < s1 && 0 < m1, {s1, m1}, 
    Integers]; 
  dat2 = FindInstance[
    6*s2^2 + 6*s2 + 1 + (6*s2 + 1)*(m2 - 1) == n && 0 < s2 && 
     0 < m2, {s2, m2}, Integers]; 
  dat3 = FindInstance[
    6*s3^2 - 2*s3 + (6*s3 - 1)*(m3 - 1) == n && 0 < s3 && 0 < m3, {s3,
      m3}, Integers]; 
  dat4 = FindInstance[
    6*s4^2 + 2*s4 + (6*s4 + 1)*(m4 - 1) == n && 0 < s4 && 0 < m4, {s4,
      m4}, Integers]; {dat1, dat2, dat3, dat4}]
"********************************************************";

This is an example where n=1 and r=2. Since “{\rm interval}”=6r-2=10. The calculation result is as follows.

====== The calculation results are shown below. ======
n= 1
r= 2
Specified interval(6*r-2)= 10
n selected= 3
Twin prime(1)= {17,19}
Twin prime(2)= {29,31}
interval(pm2-pp1)= 10
PrimeQ[(1),(2)]= {{True,True},{True,True}}
If you want to find the next twin prime, start with n= 4

The calculation result shows the input values n=1 and r=2. It shows that r=2 to {\rm Specified\ interval} \;(6r-2) = 10. The search starts from n=1 and the solution is found with n\; {\rm selected} = 3. This shows that the two twin primes obtained as a solution are {\rm Twin \,prime}\; (1) = \{17,19\} and {\rm Twin \,prime}\; (2) = \{29,31\}. It is judged by {\rm PrimeQ}\; [(1), (2)] that the four numbers are indeed prime numbers. Finally, it indicates the next n to start when continuing the search. One search interval is in the range of n to (n+100). If no solution is obtained within the interval, n for searching beyond is also displayed here.

If r=1 here, “{\rm interval}” = 4. In this case, it will be the same as " Search \;for \;consecutive \;twin \;primes " described in the previous report.

In writing this program, I used {\rm Module}\; [] and {\rm func} [n {\text \_}]: = for the first time. Thanks to Claudio-san and William-san for teaching us.

The following table shows two calculation examples. {\rm Table} \,1 is an example of r = 1, ie “{\rm interval}” = 6r-2 = 4, and {\rm Table} \,2 is an example of r = 5, ie “{\rm interval}” = 6r-2 = 28. In both cases, the search was started from n=1, and 50 sets of the selected n and the two pairs of twin primes that were the solutions were recorded.

.

{\rm Table}\;1\;\; r=2 \;{\rm ( Interval}=10 \ )
\begin{array}{|c|c|c|} \hline n\;(r=2)& {\rm Twin \ prime\ (1)}&{\rm Twin \ prime \ (2)}\\ \hline 3 & \text{{17,19}} & \text{{29,31}} \\\ 5 & \text{{29,31}} & \text{{29,31}}\\\ 10 & \text{{59,61}} & \text{{71,73}}\\\ 23 & \text{{137,139}} & \text{{149,151}}\\\ 30 & \text{{179,181}} & \text{{191,193}}\\\ 38 & \text{{227,229}} & \text{{239,241}}\\\ 45 & \text{{269,271}} & \text{{281,283}}\\\ 70 & \text{{419,421}} & \text{{431,433}}\\\ 135 & \text{{809,811}} & \text{{821,823}}\\\ 170 & \text{{1019,1021}} & \text{{1031,1033}}\\\ 175 & \text{{1049,1051}} & \text{{1061,1063}}\\\ 213 & \text{{1277,1279}} & \text{{1289,1291}}\\\ 215 & \text{{1289,1291}} & \text{{1301,1303}}\\\ 268 & \text{{1607,1609}} & \text{{1619,1621}}\\\ 355 & \text{{2129,2131}} & \text{{2141,2143}}\\\ 465 & \text{{2789,2791}} & \text{{2801,2803}}\\\ 560 & \text{{3359,3361}} & \text{{3371,3373}}\\\ 588 & \text{{3527,3529}} & \text{{3539,3541}}\\\ 703 & \text{{4217,4219}} & \text{{4229,4231}}\\\ 705 & \text{{4229,4231}} & \text{{4241,4243}}\\\ 710 & \text{{4259,4261}} & \text{{4271,4273}}\\\ 773 & \text{{4637,4639}} & \text{{4649,4651}}\\\ 798 & \text{{4787,4789}} & \text{{4799,4801}}\\\ 835 & \text{{5009,5011}} & \text{{5021,5023}}\\\ 940 & \text{{5639,5641}} & \text{{5651,5653}}\\\ 978 & \text{{5867,5869}} & \text{{5879,5881}}\\\ 1115 & \text{{6689,6691}} & \text{{6701,6703}}\\\ 1130 & \text{{6779,6781}} & \text{{6791,6793}}\\\ 1158 & \text{{6947,6949}} & \text{{6959,6961}}\\\ 1258 & \text{{7547,7549}} & \text{{7559,7561}}\\\ 1370 & \text{{8219,8221}} & \text{{8231,8233}}\\\ 1500 & \text{{8999,9001}} & \text{{9011,9013}}\\\ 1570 & \text{{9419,9421}} & \text{{9431,9433}}\\\ 1843 & \text{{11057,11059}} & \text{{11069,11071}}\\\ 1860 & \text{{11159,11161}} & \text{{11171,11173}}\\\ 2040 & \text{{12239,12241}} & \text{{12251,12253}}\\\ 2280 & \text{{13679,13681}} & \text{{13691,13693}}\\\ 2285 & \text{{13709,13711}} & \text{{13721,13723}}\\\ 2333 & \text{{13997,13999}} & \text{{14009,14011}}\\\ 2425 & \text{{14549,14551}} & \text{{14561,14563}}\\\ 2985 & \text{{17909,17911}} & \text{{17921,17923}}\\\ 3008 & \text{{18047,18049}} & \text{{18059,18061}}\\\ 3020 & \text{{18119,18121}} & \text{{18131,18133}}\\\ 3598 & \text{{21587,21589}} & \text{{21599,21601}}\\\ 3600 & \text{{21599,21601}} & \text{{21611,21613}}\\\ 3838 & \text{{23027,23029}} & \text{{23039,23041}}\\\ 4375 & \text{{26249,26251}} & \text{{26261,26263}}\\\ 4450 & \text{{26699,26701}} & \text{{26711,26713}}\\\ 4480 & \text{{26879,26881}} & \text{{26891,26893}}\\\ 4588 & \text{{27527,27529}} & \text{{27539,27541}}\\\ \text{Omitted below} & \text{Omitted below} &\text{Omitted below}\\ \hline \end{array}

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{\rm Table}\;2\;\; r=5 \;{\rm ( Interval}=28 \ )
\begin{array}{|c|c|c|} \hline n\;(r=5)& {\rm Twin \ prime\ (1)}&{\rm Twin \ prime \ (2)}\\ \hline 12 & \text{{71,73}} & \text{{101,103}}\\\ 18 & \text{{107,109}} & \text{{137,139}}\\\ 25 & \text{{149,151}} & \text{{179,181}}\\\ 33 & \text{{197,199}} & \text{{227,229}}\\\ 40 & \text{{239,241}} & \text{{269,271}}\\\ 47 & \text{{281,283}} & \text{{311,313}}\\\ 72 & \text{{431,433}} & \text{{461,463}}\\\ 95 & \text{{569,571}} & \text{{599,601}}\\\ 138 & \text{{827,829}} & \text{{857,859}}\\\ 177 & \text{{1061,1063}} & \text{{1091,1093}}\\\ 242 & \text{{1451,1453}} & \text{{1481,1483}}\\\ 278 & \text{{1667,1669}} & \text{{1697,1699}}\\\ 333 & \text{{1997,1999}} & \text{{2027,2029}}\\\ 373 & \text{{2237,2239}} & \text{{2267,2269}}\\\ 385 & \text{{2309,2311}} & \text{{2339,2341}}\\\ 443 & \text{{2657,2659}} & \text{{2687,2689}}\\\ 495 & \text{{2969,2971}} & \text{{2999,3001}}\\\ 550 & \text{{3299,3301}} & \text{{3329,3331}}\\\ 555 & \text{{3329,3331}} & \text{{3359,3361}}\\\ 637 & \text{{3821,3823}} & \text{{3851,3853}}\\\ 670 & \text{{4019,4021}} & \text{{4049,4051}}\\\ 688 & \text{{4127,4129}} & \text{{4157,4159}}\\\ 753 & \text{{4517,4519}} & \text{{4547,4549}}\\\ 1045 & \text{{6269,6271}} & \text{{6299,6301}}\\\ 1110 & \text{{6659,6661}} & \text{{6689,6691}}\\\ 1243 & \text{{7457,7459}} & \text{{7487,7489}}\\\ 1260 & \text{{7559,7561}} & \text{{7589,7591}}\\\ 1433 & \text{{8597,8599}} & \text{{8627,8629}}\\\ 1495 & \text{{8969,8971}} & \text{{8999,9001}}\\\ 1502 & \text{{9011,9013}} & \text{{9041,9043}}\\\ 1668 & \text{{10007,10009}} & \text{{10037,10039}}\\\ 1673 & \text{{10037,10039}} & \text{{10067,10069}}\\\ 1712 & \text{{10271,10273}} & \text{{10301,10303}}\\\ 1717 & \text{{10301,10303}} & \text{{10331,10333}}\\\ 1738 & \text{{10427,10429}} & \text{{10457,10459}}\\\ 1750 & \text{{10499,10501}} & \text{{10529,10531}}\\\ 1810 & \text{{10859,10861}} & \text{{10889,10891}}\\\ 1990 & \text{{11939,11941}} & \text{{11969,11971}}\\\ 2007 & \text{{12041,12043}} & \text{{12071,12073}}\\\ 2317 & \text{{13901,13903}} & \text{{13931,13933}}\\\ 2427 & \text{{14561,14563}} & \text{{14591,14593}}\\\ 2555 & \text{{15329,15331}} & \text{{15359,15361}}\\\ 2898 & \text{{17387,17389}} & \text{{17417,17419}}\\\ 2993 & \text{{17957,17959}} & \text{{17987,17989}}\\\ 3197 & \text{{19181,19183}} & \text{{19211,19213}}\\\ 3327 & \text{{19961,19963}} & \text{{19991,19993}}\\\ 3332 & \text{{19991,19993}} & \text{{20021,20023}}\\\ 3413 & \text{{20477,20479}} & \text{{20507,20509}}\\\ 3453 & \text{{20717,20719}} & \text{{20747,20749}}\\\ 3497 & \text{{20981,20983}} & \text{{21011,21013}}\\\ \text{Omitted below} & \text{Omitted below} &\text{Omitted below}\\ \hline \end{array}
2 Likes
2

The text below the graph did not display a few lines correctly, so I’ll take this opportunity to show you what I intended.

Why was it displayed correctly in “Post Preview”?

-------Correct text

The horizontal axis of this graph is the “{\rm interval}” (=p_{m2}-p_{p1}) of the twin prime Tp_1 and the next twin prime Tp_2. The vertical axis is the frequency of occurrence, and the parameter is approximately 61000. What you can see in the graph is the fact that there is a solution only if the “{\rm interval}” is 4,10,16,22,?. In the case of 6,8,12,14,? there is no solution. I will omit the explanation, but this is because one or both of Tp_2 is not a prime number depending on the condition of equation (1). (the sum of odd and odd numbers is even, multiples of 3 are involved)

3

Since it was not displayed correctly, a sentence excluding processing is presented. Please read subscripts by yourself.

-------Correct text

The horizontal axis of this graph is the " interval" (=p_m2-p_p1) of the twin prime Tp_1 and the next twin prime Tp_2. The vertical axis is the frequency of occurrence, and the parameter is approximately 61000. What you can see in the graph is the fact that there is a solution only if the " interval" is 4,10,16,22,?. In the case of 6,8,12,14,? there is no solution. I will omit the explanation, but this is because one or both of Tp_2 is not a prime number depending on the condition of equation (1). (the sum of odd and odd numbers is even, multiples of 3 are involved)