Find prime number closest to any integer (upper side)

Ideas
1

Here is a program that searches for a prime number closest to an arbitrary integer.? However, there are conditions. The condition is that the specified integer is 5 or more and the search is limited to the upper side.

Prime numbers greater than 5 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}

Equations (1{\rm a}) and (1{\rm b}) determine prime numbers of type (6n-1), and equations (1{\rm c}) and (1{\rm d}) determine prime numbers of type (6n + 1). Given an arbitrary positive integer n in equation (1) and there are no integer solutions in s and m, n determines the prime number. If there is no integer solution for s and m in equations (1{\rm a}) and (1{\rm b}), let n be n_m. Similarly, in equations (1{\rm c}) and (1{\rm d}), n is expressed as n_p when there is no integer solution for s and m. The prime number is determined from n_m and n_p according to equation (2).

\begin{eqnarray} (6n-1)\text{ type prime}&=&6n_m-1\tag{2a} \\ (6n+1)\text{ type prime}&=&6n_p+1\tag{2b} \\ \end{eqnarray}
\begin{equation}n_m\ {\rm and}{\text{ }}n_p: {\rm In\ Equation\ (1),{\it n}\ when\ there\ is\ no\ integer\ solution\ for\ {\it s}\ and\ {\it m}.}\end{equation}

Here is a program that uses this equation (1) and (2) to find the nearest prime number to an arbitrarily specified positive integer {\rm j}. There is a relation n=(\;{\rm j}-1)\,/\,6\; between {\rm j} and n. I wrote this in Mathematica as n = {\rm Quotient}\, [(\,{\rm j}-1), 6]\, and converted {\rm j} to n. If any integer is input to {\rm j}, the prime number closest to {\rm j} is output. Search the range of n to (n + 100).

j = 5;

Clear[nm, np, pm, pp, minp];
n = Quotient[(j - 1), 6];
If[j >= 5 && n >= 0 && IntegerQ[j],
  Do[{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]};
   
   If[dat1 == {} && dat2 == {}, nm = n, nm = 0]; 
   If[dat1 == {} && dat2 == {} && ((6*nm - 1) - j) >= 0, Break[]], {n,
     n, n + 100}];
  
  Do[{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]};
   
   If[dat3 == {} && dat4 == {}, np = n, np = 0]; 
   If[dat3 == {} && dat4 == {} && ((6*np + 1) - j) >= 0, Break[]], {n,
     n, n + 100}];
  
  Print["====== The calculation results are shown below. ======"];
  
  If[nm > 0, pm = 6*nm - 1, pm = 0];
  If[np > 0, pp = 6*np + 1, pp = 0];
  If[pm >= 1 && pp >= 1, minp = Min[pm, pp], minp = Max[pm, pp]];
  
  Print["j = ", j];
  If[Or[pm >= 1, pp >= 1], 
   Print["Prime number closest to j (upper) = " , minp], 
   Print["Prime number closest to j (upper)= There is no solution in \
the searched interval."]];
  
  If[Or[nm >= 1, np >= 1], Print["difference = ", minp - j], 
   Print["difference = -"]];
  
  If[Or[nm >= 1, np >= 1], Print["PrimeQ = ", PrimeQ[minp]], 
   Print["PrimeQ = -"]];
  
  If[Or[pm >= 1, pp >= 1], Print["Enter j for the next search. "], 
   Print["There is no solution for this interval. If you want to \
continue, start with the next j. j = ", (n + 101)*6]], 
  Print["***** ! Please enter an integer greater than 5 *****"]];

The calculation result is shown as follows. This is an example when {\rm j} = 5.

====== The calculation results are shown below. ======
j = 5
Prime number closest to j (upper) = 5
difference = 0
PrimeQ = True
Enter j for the next search.

The closest prime number for {\rm j} = 5 is 5, so the difference is difference = 0. {\rm PrimeQ} confirms that it is a prime number.

Let’s put a little bigger number. Let’s assume {\rm j} = 78. Does “78” have any meaning? Nothing in particular. This is my age. The calculation result is as follows.

====== The calculation results are shown below. ======
j = 78
Prime number closest to j (upper) = 79
difference = 1
PrimeQ = True
Enter j for the next search.

The closest prime number to {\rm j} = 78 is “79”. (Next year my age will be a prime number)

Let’s check a slightly larger number.
Try:

{\rm j}\,=\,26881171418161354484126255515800135873611119

This number is {\rm{Round}}\, [{\rm{E}}^{100}]. The calculation result is as follows.

====== The calculation results are shown below. ======
j = 26881171418161354484126255515800135873611119
Prime number closest to j (upper) = 26881171418161354484126255515800135873611287
difference = 168
PrimeQ = True
Enter j for the next search.

The input integer {\rm j} and the nearest prime number are as follows.

{\rm j}\,=\,26881171418161354484126255515800135873611119

{\rm A\ prime\ number\ close\ to\ j}\,=\,26881171418161354484126255515800135873611287

Some calculation examples are shown in the table.

\begin{array}{ccc} {\rm j\;\, (Any\ integer)} & {\rm prime\ number\ closest\ to\ j\ (upper)}&{\rm difference}\\ 5 & 5 &0\\ 10 & 11&1 \\ 15 & 17&2 \\ 20 & 23&3 \\ 78 & 79&1 \\ 123 & 127&4 \\ 567 & 569&2 \\ 999 & 1009&10 \\ 54321 & 54323&2 \\ 123456789 & 123456791 &2\\ 5454545454545454 & 5454545454545483&29 \\ 9876543210123456789 & 9876543210123456803&14 \\ 3141592653589793238462643 & 3141592653589793238462773&130 \\ \text{Omitted below} & \text{Omitted below} &\text{Omitted below} \end{array}
2 Likes
2

Or just use NextPrime[x] :slight_smile:

3

I don’t know who you are, but I would like to thank you for improving my program that I posted two years ago.

You defined j, and used Quotient[] and FindInstance[] to find the solution.

I didn’t know about Quotient[] and FindInstance[], so this will be helpful for me in the future.

… I’m getting old. I’m getting older, and the months are passing quickly, and I feel like difference>2.