# Twin primes greater than 5 (2)

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 When $n$ is a positive integer and both ( $6n-1$) and ( $6n+1$) are prime numbers, they are twin prime numbers. ( $6n-1$) is an infinite sequence of { $5,11,17,23, ?$}, and ( $6n+1$) is an infinite sequence of { $7,13,19,25, ?$}. An integer belonging to the sequence ( $6n-1$) is denoted as $n_m$, and an integer belonging to ( $6n+1$) is denoted as $n_p$. However, during the discussion, all positive integers are included in $n_m$ and $n_p$. Integers that do not belong to ( $6n?1$) are removed inside the program. Twin prime numbers have a relationship of $n_p=n_m+2$.Equation ( $1$) determines whether $n_m$ and $n_p$ are twin primes or not. Give $n_m$ and $n_p$ to the right side of equation ( $1$), and give an arbitrary integer $s$. If $m$ has any integer solution, $n_m$ and $n_p$ are not twin primes. If there is no integer solution for $m$ , $n_m$ and $n_p$ are twin primes. $$\begin{eqnarray} m_1&=&\frac{n_m-5+36s_1-36s_1^2}{6\cdot(6s_1-1)}\tag{1a} \\ m_2&=&\frac{n_m+1-36s_2^2}{6\cdot(6s_2+1)}\tag{1b} \\ m_3&=&\frac{n_p-7+48s_3-36s_3^2}{6\cdot(6s_3-1)}\tag{1c}\\ m_4&=&\frac{n_p+5+24s_4-36s_4^2}{6\cdot(6s_4+1)}\tag{1d} \end{eqnarray}$$ \begin{align*} & s=1,2,3\cdots \\ & n_m=(6n-1) \\ & n_p=(6n+1)=n_m+2 \\ & n=1,2,3\cdots \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ \end{align*} $$$$m= \left \{ \begin{array}{l} {\text {If there is at least one integer solution : }}n_m{\text { and }}n_p{\text { is not twin prime}} \; \\ {\text {If there is no integer solution : }}n_m{\text { and }}n_p{\text { is a twin prime number}} \end{array} \right.\\$$$$ $s$ starts searching from $s=1$ and must search in step $1$ up to $s_{max}$. $s_{max}$ is determined depending on $n_p$. Equation ( $2$) was used to calculate $s_{max}$. $$\begin{eqnarray} s_{max}=\frac{\sqrt{n_p}}{6}+1\tag{2}\\ \end{eqnarray}$$ \begin{align*} &s_{max} :{\text {Starting from }s=1{\text {, the upper limit of the search range (integer value) }}} \\ &n_p \text{ : Positive integer that tests twin primes = (6n + 1) = (6n-1)+2}\\ \end{align*}Note that $n_m$ and $n_p$ are given as arbitrary integers, and only those belonging to ( $6n?1$) in the program are selected.The program is shown below. nmstart = 1; nmend = 1000; "**********************************************"; nm = nmstart; np = nm + 2; nf = (nm + 1)/6; smax = IntegerPart[Sqrt[np]/6] + 1; Clear[f1, f2, f3, m1, m2, m3, m4]; Print["====== The calculation results are shown below. ======"]; func1[s_] := (nm - 5 + 36*s - 36*s^2)/(-6 + 36*s); func2[s_] := (nm + 1 - 36*s^2)/(6 + 36*s); func3[s_] := (np - 7 + 48*s - 36*s^2)/(-6 + 36*s); func4[s_] := (np + 5 + 24*s - 36*s^2)/(6 + 36*s); If[IntegerQ[nm] && nm > 0 && nmend > nmstart, Print["nmstart= ", nm, "\nnmend= ", nmend]; Print["----------------"]; Do[ np = nm + 2; smax = IntegerPart[Sqrt[np]/6] + 1; nf = (nm + 1)/6; f1 = 0; f2 = 0; Do[If[IntegerQ[func1[s]] && func1[s] > 0, m1 = 1, m1 = 0]; If[IntegerQ[func2[s]] && func2[s] > 0, m2 = 1, m2 = 0]; If[IntegerQ[func3[s]] && func3[s] > 0, m3 = 1, m3 = 0]; If[IntegerQ[func4[s]] && func4[s] > 0, m4 = 1, m4 = 0]; f1 = f1 + m1 + m2; f2 = f2 + m3 + m4; f3 = If[IntegerQ[nf], 1, 0]; If[s == smax, Break[]], {s, 1, smax}]; If[f1 == 0 && f2 == 0 && f3 == 1, Print["Twin prime.\n", nm " is a prime number of (6n-1) type \n", np " is a prime number of (6n+1) type "]]; If[f1 == 0 && f2 == 0 && f3 == 1, Print["PrimeQ[", nm, "]= ", PrimeQ[nm]]; Print["PrimeQ[", np, "]= ", PrimeQ[np], "\n"]]; If[nm == nmend, Break[]], {nm, nmstart, nmend}]; If[f1 == 0 && f2 == 0 && f3 == 1, Print["Twin prime.\n", nm " is a prime number of (6n-1) type \n", np " is a prime number of (6n+1) type"]]; If[f1 == 0 && f2 == 0 && f3 == 1, Print["PrimeQ[", nm, "]= ", PrimeQ[nm]]; Print["PrimeQ[", np, "]= ", PrimeQ[np "\n"]]], Print["!! The condition has not been met."]]; This is an example in which the input values are $n_m start=1$ and $n_m end=1000$. The calculation result is as follows. ====== The calculation results are shown below. ====== nmstart= 1 nmend= 1000 ---------------- Twin prime. 5 is a prime number of (6n-1) type 7 is a prime number of (6n+1) type PrimeQ[5]= True PrimeQ[7]= True Twin prime. 11 is a prime number of (6n-1) type 13 is a prime number of (6n+1) type PrimeQ[11]= True PrimeQ[13]= True Twin prime. 17 is a prime number of (6n-1) type 19 is a prime number of (6n+1) type PrimeQ[17]= True PrimeQ[19]= True Twin prime. 29 is a prime number of (6n-1) type 31 is a prime number of (6n+1) type PrimeQ[29]= True PrimeQ[31]= True Twin prime. 41 is a prime number of (6n-1) type 43 is a prime number of (6n+1) type PrimeQ[41]= True PrimeQ[43]= True Twin prime. 59 is a prime number of (6n-1) type 61 is a prime number of (6n+1) type PrimeQ[59]= True PrimeQ[61]= True Twin prime. 71 is a prime number of (6n-1) type 73 is a prime number of (6n+1) type PrimeQ[71]= True PrimeQ[73]= True Twin prime. 101 is a prime number of (6n-1) type 103 is a prime number of (6n+1) type PrimeQ[101]= True PrimeQ[103]= True Twin prime. 107 is a prime number of (6n-1) type 109 is a prime number of (6n+1) type PrimeQ[107]= True PrimeQ[109]= True Twin prime. 137 is a prime number of (6n-1) type 139 is a prime number of (6n+1) type PrimeQ[137]= True PrimeQ[139]= True Twin prime. 149 is a prime number of (6n-1) type 151 is a prime number of (6n+1) type PrimeQ[149]= True PrimeQ[151]= True Twin prime. 179 is a prime number of (6n-1) type 181 is a prime number of (6n+1) type PrimeQ[179]= True PrimeQ[181]= True Twin prime. 191 is a prime number of (6n-1) type 193 is a prime number of (6n+1) type PrimeQ[191]= True PrimeQ[193]= True Twin prime. 197 is a prime number of (6n-1) type 199 is a prime number of (6n+1) type PrimeQ[197]= True PrimeQ[199]= True Twin prime. 227 is a prime number of (6n-1) type 229 is a prime number of (6n+1) type PrimeQ[227]= True PrimeQ[229]= True Twin prime. 239 is a prime number of (6n-1) type 241 is a prime number of (6n+1) type PrimeQ[239]= True PrimeQ[241]= True Twin prime. 269 is a prime number of (6n-1) type 271 is a prime number of (6n+1) type PrimeQ[269]= True PrimeQ[271]= True Twin prime. 281 is a prime number of (6n-1) type 283 is a prime number of (6n+1) type PrimeQ[281]= True PrimeQ[283]= True Twin prime. 311 is a prime number of (6n-1) type 313 is a prime number of (6n+1) type PrimeQ[311]= True PrimeQ[313]= True Twin prime. 347 is a prime number of (6n-1) type 349 is a prime number of (6n+1) type PrimeQ[347]= True PrimeQ[349]= True Twin prime. 419 is a prime number of (6n-1) type 421 is a prime number of (6n+1) type PrimeQ[419]= True PrimeQ[421]= True Twin prime. 431 is a prime number of (6n-1) type 433 is a prime number of (6n+1) type PrimeQ[431]= True PrimeQ[433]= True Twin prime. 461 is a prime number of (6n-1) type 463 is a prime number of (6n+1) type PrimeQ[461]= True PrimeQ[463]= True Twin prime. 521 is a prime number of (6n-1) type 523 is a prime number of (6n+1) type PrimeQ[521]= True PrimeQ[523]= True Twin prime. 569 is a prime number of (6n-1) type 571 is a prime number of (6n+1) type PrimeQ[569]= True PrimeQ[571]= True Twin prime. 599 is a prime number of (6n-1) type 601 is a prime number of (6n+1) type PrimeQ[599]= True PrimeQ[601]= True Twin prime. 617 is a prime number of (6n-1) type 619 is a prime number of (6n+1) type PrimeQ[617]= True PrimeQ[619]= True Twin prime. 641 is a prime number of (6n-1) type 643 is a prime number of (6n+1) type PrimeQ[641]= True PrimeQ[643]= True Twin prime. 659 is a prime number of (6n-1) type 661 is a prime number of (6n+1) type PrimeQ[659]= True PrimeQ[661]= True Twin prime. 809 is a prime number of (6n-1) type 811 is a prime number of (6n+1) type PrimeQ[809]= True PrimeQ[811]= True Twin prime. 821 is a prime number of (6n-1) type 823 is a prime number of (6n+1) type PrimeQ[821]= True PrimeQ[823]= True Twin prime. 827 is a prime number of (6n-1) type 829 is a prime number of (6n+1) type PrimeQ[827]= True PrimeQ[829]= True Twin prime. 857 is a prime number of (6n-1) type 859 is a prime number of (6n+1) type PrimeQ[857]= True PrimeQ[859]= True Twin prime. 881 is a prime number of (6n-1) type 883 is a prime number of (6n+1) type PrimeQ[881]= True PrimeQ[883]= True In the display of the calculation result, $n_m start$ and $n_m end$ are shown first. Subsequently, the selected twin prime numbers and their types are displayed. Confirm that it is a prime number with ${\rm PrimeQ}$ [].There were $34$ pairs of twin primes in the section where $n_m$ was $1$ to $1000$. How many twin primes are there between $1001$ and$2000$? The following graph plots the number of twin primes included in the section width $1000$.At the start, the twin prime numbers tend to decrease, but as the twin prime numbers increase, the decreasing trend appears to be slow. $--------$There are rules for twin primes greater than $5$.? The end of $n_m$ is limited to [ $1,7,9$ ].? The end of $n_p$ is limited to [ $3,9,1$ ].This rule can be understood with a little thought. This rule can be used to eliminate unnecessary calculations, but it has not been applied this time.
 The fourth line from the end was incorrect, so correct it as follows:?8th January 2020 $--------$There are rules for twin primes with more than two digits.? The end of $n_m$ is limited to [ $1,7,9$ ].? The end of $n_p$ is limited to [ $3,9,1$ ].This rule can be understood with a little thought. This rule can be used to eliminate unnecessary calculations, but it has not been applied this time.