View source for Earliest difference between prime gaps

{{task}}

When calculating prime numbers > 2, the difference between adjacent primes is always an even number. This difference, also referred to as the gap, varies in an random pattern; at least, no pattern has ever been discovered, and it is strongly conjectured that no pattern exists. However, it is also conjectured that between some two adjacent primes will be a gap corresponding to every positive even integer.


<div style="float:left;padding-left:2em;padding-right:3em;">
{|class="wikitable"
!gap!!minimal<br>starting<br>prime!!ending<br>prime
|-
|2||3||5
|-
|4||7||11
|-
|6||23||29
|-
|8||89||97
|-
|10||139||149
|-
|12||199||211
|-
|14||113||127
|-
|16||1831||1847
|-
|18||523||541
|-
|20||887||907
|-
|22||1129||1151
|-
|24||1669||1693
|-
|26||2477||2503
|-
|28||2971||2999
|-
|30||4297||4327
|}
</div>
This task involves locating the minimal primes corresponding to those gaps.

Though every gap value exists, they don't seem to come in any particular order. For example, this table shows the gaps and minimum starting value primes for 2 through 30:


For the purposes of this task, considering only primes greater than 2, consider prime gaps that differ by exactly two to be adjacent.



;Task

For each order of magnitude '''m''' from '''10¹''' through '''10⁶''':

* Find the first two sets of minimum adjacent prime gaps where the absolute value of the difference between the prime gap start values is greater than '''m'''.


;E.G.

For an '''m''' of '''10¹'''; 

The start value of gap 2 is 3, the start value of gap 4 is 7, the difference is 7 - 3 or 4. 4 < '''10¹''' so keep going.

The start value of gap 4 is 7, the start value of gap 6 is 23, the difference is 23 - 7, or 16. 16 > '''10¹''' so this the earliest adjacent gap difference > '''10¹'''.


;Stretch goal

* Do the same for '''10⁷''' and '''10⁸''' (and higher?) orders of magnitude

Note: the earliest value found for each order of magnitude may not be unique, in fact, ''is'' not unique; also, with the gaps in ascending order, the minimal starting values are not strictly ascending.

=={{header|C++}}==
{{libheader|Primesieve}}
<syntaxhighlight lang="cpp">#include <iostream>
#include <locale>
#include <unordered_map>

#include <primesieve.hpp>

class prime_gaps {
public:
    prime_gaps() { last_prime_ = iterator_.next_prime(); }
    uint64_t find_gap_start(uint64_t gap);
private:
    primesieve::iterator iterator_;
    uint64_t last_prime_;
    std::unordered_map<uint64_t, uint64_t> gap_starts_;
};

uint64_t prime_gaps::find_gap_start(uint64_t gap) {
    auto i = gap_starts_.find(gap);
    if (i != gap_starts_.end())
        return i->second;
    for (;;) {
        uint64_t prev = last_prime_;
        last_prime_ = iterator_.next_prime();
        uint64_t diff = last_prime_ - prev;
        gap_starts_.emplace(diff, prev);
        if (gap == diff)
            return prev;
    }
}

int main() {
    std::cout.imbue(std::locale(""));
    const uint64_t limit = 100000000000;
    prime_gaps pg;
    for (uint64_t pm = 10, gap1 = 2;;) {
        uint64_t start1 = pg.find_gap_start(gap1);
        uint64_t gap2 = gap1 + 2;
        uint64_t start2 = pg.find_gap_start(gap2);
        uint64_t diff = start2 > start1 ? start2 - start1 : start1 - start2;
        if (diff > pm) {
            std::cout << "Earliest difference > " << pm
                      << " between adjacent prime gap starting primes:\n"
                      << "Gap " << gap1 << " starts at " << start1 << ", gap "
                      << gap2 << " starts at " << start2 << ", difference is "
                      << diff << ".\n\n";
            if (pm == limit)
                break;
            pm *= 10;
        } else {
            gap1 = gap2;
        }
    }
}</syntaxhighlight>

{{out}}
<pre>
Earliest difference > 10 between adjacent prime gap starting primes:
Gap 4 starts at 7, gap 6 starts at 23, difference is 16.

Earliest difference > 100 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1,831, difference is 1,718.

Earliest difference > 1,000 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1,831, difference is 1,718.

Earliest difference > 10,000 between adjacent prime gap starting primes:
Gap 36 starts at 9,551, gap 38 starts at 30,593, difference is 21,042.

Earliest difference > 100,000 between adjacent prime gap starting primes:
Gap 70 starts at 173,359, gap 72 starts at 31,397, difference is 141,962.

Earliest difference > 1,000,000 between adjacent prime gap starting primes:
Gap 100 starts at 396,733, gap 102 starts at 1,444,309, difference is 1,047,576.

Earliest difference > 10,000,000 between adjacent prime gap starting primes:
Gap 148 starts at 2,010,733, gap 150 starts at 13,626,257, difference is 11,615,524.

Earliest difference > 100,000,000 between adjacent prime gap starting primes:
Gap 198 starts at 46,006,769, gap 200 starts at 378,043,979, difference is 332,037,210.

Earliest difference > 1,000,000,000 between adjacent prime gap starting primes:
Gap 276 starts at 649,580,171, gap 278 starts at 4,260,928,601, difference is 3,611,348,430.

Earliest difference > 10,000,000,000 between adjacent prime gap starting primes:
Gap 332 starts at 5,893,180,121, gap 334 starts at 30,827,138,509, difference is 24,933,958,388.

Earliest difference > 100,000,000,000 between adjacent prime gap starting primes:
Gap 386 starts at 35,238,645,587, gap 388 starts at 156,798,792,223, difference is 121,560,146,636.

</pre>

=={{header|F_Sharp|F#}}==
This task uses [http://www.rosettacode.org/wiki/Extensible_prime_generator#The_functions Extensible Prime Generator (F#)]
<syntaxhighlight lang="fsharp">
// Earliest difference between prime gaps. Nigel Galloway: December 1st., 2021
let fN y=let i=System.Collections.Generic.SortedDictionary<int64,int64>()
         let fN()=i|>Seq.pairwise|>Seq.takeWhile(fun(n,g)->g.Key=n.Key+2L)|>Seq.tryFind(fun(n,g)->abs(n.Value-g.Value)>y)
         (fun(n,g)->let e=g-n in match i.TryGetValue(e) with (false,_)->i.Add(e,n); fN() |_->None)
[1..9]|>List.iter(fun g->let fN=fN(pown 10 g) in let n,i=(primes64()|>Seq.skip 1|>Seq.pairwise|>Seq.map fN|>Seq.find Option.isSome).Value
                         printfn $"%10d{pown 10 g} -> distance between start of gap %d{n.Key}=%d{n.Value} and start of gap %d{i.Key}=%d{i.Value} is %d{abs((n.Value)-(i.Value))}")
</syntaxhighlight>
{{out}}
<pre>
        10 -> distance between start of gap 4=7 and start of gap 6=23 is 16
       100 -> distance between start of gap 14=113 and start of gap 16=1831 is 1718
      1000 -> distance between start of gap 14=113 and start of gap 16=1831 is 1718
     10000 -> distance between start of gap 36=9551 and start of gap 38=30593 is 21042
    100000 -> distance between start of gap 70=173359 and start of gap 72=31397 is 141962
   1000000 -> distance between start of gap 100=396733 and start of gap 102=1444309 is 1047576
  10000000 -> distance between start of gap 148=2010733 and start of gap 150=13626257 is 11615524
 100000000 -> distance between start of gap 198=46006769 and start of gap 200=378043979 is 332037210
1000000000 -> distance between start of gap 276=649580171 and start of gap 278=4260928601 is 3611348430
</pre>
=={{header|Go}}==
{{trans|Wren}}
{{libheader|Go-rcu}}
<syntaxhighlight lang="go">package main

import (
    "fmt"
    "rcu"
)

func main() {
    limit := int(1e9)
    gapStarts := make(map[int]int)
    primes := rcu.Primes(limit * 5)
    for i := 1; i < len(primes); i++ {
        gap := primes[i] - primes[i-1]
        if _, ok := gapStarts[gap]; !ok {
            gapStarts[gap] = primes[i-1]
        }
    }
    pm := 10
    gap1 := 2
    for {
        for _, ok := gapStarts[gap1]; !ok; {
            gap1 += 2
        }
        start1 := gapStarts[gap1]
        gap2 := gap1 + 2
        if _, ok := gapStarts[gap2]; !ok {
            gap1 = gap2 + 2
            continue
        }
        start2 := gapStarts[gap2]
        diff := start2 - start1
        if diff < 0 {
            diff = -diff
        }
        if diff > pm {
            cpm := rcu.Commatize(pm)
            cst1 := rcu.Commatize(start1)
            cst2 := rcu.Commatize(start2)
            cd := rcu.Commatize(diff)
            fmt.Printf("Earliest difference > %s between adjacent prime gap starting primes:\n", cpm)
            fmt.Printf("Gap %d starts at %s, gap %d starts at %s, difference is %s.\n\n", gap1, cst1, gap2, cst2, cd)
            if pm == limit {
                break
            }
            pm *= 10
        } else {
            gap1 = gap2
        }
    }
}</syntaxhighlight>

{{out}}
<pre>
Earliest difference > 10 between adjacent prime gap starting primes:
Gap 4 starts at 7, gap 6 starts at 23, difference is 16.

Earliest difference > 100 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1,831, difference is 1,718.

Earliest difference > 1,000 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1,831, difference is 1,718.

Earliest difference > 10,000 between adjacent prime gap starting primes:
Gap 36 starts at 9,551, gap 38 starts at 30,593, difference is 21,042.

Earliest difference > 100,000 between adjacent prime gap starting primes:
Gap 70 starts at 173,359, gap 72 starts at 31,397, difference is 141,962.

Earliest difference > 1,000,000 between adjacent prime gap starting primes:
Gap 100 starts at 396,733, gap 102 starts at 1,444,309, difference is 1,047,576.

Earliest difference > 10,000,000 between adjacent prime gap starting primes:
Gap 148 starts at 2,010,733, gap 150 starts at 13,626,257, difference is 11,615,524.

Earliest difference > 100,000,000 between adjacent prime gap starting primes:
Gap 198 starts at 46,006,769, gap 200 starts at 378,043,979, difference is 332,037,210.

Earliest difference > 1,000,000,000 between adjacent prime gap starting primes:
Gap 276 starts at 649,580,171, gap 278 starts at 4,260,928,601, difference is 3,611,348,430.
</pre>

=={{header|J}}==

Implementation: 

<syntaxhighlight lang="j">lowpgap=: {{
  magnitude=. ref=. 10^y
  whilst. -.1 e. ok do.
    magnitude=. 10*magnitude
    g=. 2-~/\ p=.p: i.magnitude
    mag=. p{~}.g i. i.&.-: >./g NB. minimum adjacent gaps
    dif=. 2-~/\ mag
    ok=. ref < dif
  end.
  (0 1+ok i.1){mag
}}</syntaxhighlight>

Task examples:

<syntaxhighlight lang="j">   lowpgap 1
7 23
   lowpgap 2
113 1831
   lowpgap 3
113 1831
   lowpgap 4
9551 30593
   lowpgap 5
31397 404597
   lowpgap 6
396733 1444309
   lowpgap 7
2010733 13626257</syntaxhighlight>

=={{header|Java}}==
Uses the PrimeGenerator class from [[Extensible prime generator#Java]].
<syntaxhighlight lang="java">import java.util.HashMap;
import java.util.Map;

public class PrimeGaps {
    private Map<Integer, Integer> gapStarts = new HashMap<>();
    private int lastPrime;
    private PrimeGenerator primeGenerator = new PrimeGenerator(1000, 500000);

    public static void main(String[] args) {
        final int limit = 100000000;
        PrimeGaps pg = new PrimeGaps();
        for (int pm = 10, gap1 = 2;;) {
            int start1 = pg.findGapStart(gap1);
            int gap2 = gap1 + 2;
            int start2 = pg.findGapStart(gap2);
            int diff = start2 > start1 ? start2 - start1 : start1 - start2;
            if (diff > pm) {
                System.out.printf(
                    "Earliest difference > %,d between adjacent prime gap starting primes:\n"
                    + "Gap %,d starts at %,d, gap %,d starts at %,d, difference is %,d.\n\n",
                    pm, gap1, start1, gap2, start2, diff);
                if (pm == limit)
                    break;
                pm *= 10;
            } else {
                gap1 = gap2;
            }
        }
    }

    private int findGapStart(int gap) {
        Integer start = gapStarts.get(gap);
        if (start != null)
            return start;
        for (;;) {
            int prev = lastPrime;
            lastPrime = primeGenerator.nextPrime();
            int diff = lastPrime - prev;
            gapStarts.putIfAbsent(diff, prev);
            if (diff == gap)
                return prev;
        }
    }
}</syntaxhighlight>

{{out}}
<pre>
Earliest difference > 10 between adjacent prime gap starting primes:
Gap 4 starts at 7, gap 6 starts at 23, difference is 16.

Earliest difference > 100 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1,831, difference is 1,718.

Earliest difference > 1,000 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1,831, difference is 1,718.

Earliest difference > 10,000 between adjacent prime gap starting primes:
Gap 36 starts at 9,551, gap 38 starts at 30,593, difference is 21,042.

Earliest difference > 100,000 between adjacent prime gap starting primes:
Gap 70 starts at 173,359, gap 72 starts at 31,397, difference is 141,962.

Earliest difference > 1,000,000 between adjacent prime gap starting primes:
Gap 100 starts at 396,733, gap 102 starts at 1,444,309, difference is 1,047,576.

Earliest difference > 10,000,000 between adjacent prime gap starting primes:
Gap 148 starts at 2,010,733, gap 150 starts at 13,626,257, difference is 11,615,524.

Earliest difference > 100,000,000 between adjacent prime gap starting primes:
Gap 198 starts at 46,006,769, gap 200 starts at 378,043,979, difference is 332,037,210.

</pre>

=={{header|Julia}}==
{{trans|Wren}}
To get to 10^9 correctly we need a limit multiplier of 5 in Julia, not 4.  Going up to 10^10 runs out of memory on my machine.
<syntaxhighlight lang="julia">using Formatting
using Primes

function primegaps(limit = 10^9)
    c(n) = format(n, commas=true)
    pri = primes(limit * 5)
    gapstarts = Dict{Int, Int}()
    for i in 2:length(pri)
        get!(gapstarts, pri[i] - pri[i - 1], pri[i - 1])
    end
    pm, gap1 = 10, 2
    while true
        while !haskey(gapstarts, gap1)
            gap1 += 2
        end
        start1 = gapstarts[gap1]
        gap2 = gap1 + 2
        if !haskey(gapstarts, gap2)
            gap1 = gap2 + 2
            continue
        end
        start2 = gapstarts[gap2]
        if ((diff = abs(start2 - start1)) > pm)
            println("Earliest difference > $(c(pm)) between adjacent prime gap starting primes:")
            println("Gap $gap1 starts at $(c(start1)), gap $(c(gap2)) starts at $(c(start2)), difference is $(c(diff)).\n")
            pm == limit && break
            pm *= 10
        else
            gap1 = gap2
        end
    end
end

primegaps()
</syntaxhighlight>{{out}}
<pre>
Earliest difference > 10 between adjacent prime gap starting primes:
Gap 4 starts at 7, gap 6 starts at 23, difference is 16.

Earliest difference > 100 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1,831, difference is 1,718.

Earliest difference > 1,000 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1,831, difference is 1,718.

Earliest difference > 10,000 between adjacent prime gap starting primes:
Gap 36 starts at 9,551, gap 38 starts at 30,593, difference is 21,042.

Earliest difference > 100,000 between adjacent prime gap starting primes:
Gap 70 starts at 173,359, gap 72 starts at 31,397, difference is 141,962.

Earliest difference > 1,000,000 between adjacent prime gap starting primes:
Gap 100 starts at 396,733, gap 102 starts at 1,444,309, difference is 1,047,576.

Earliest difference > 10,000,000 between adjacent prime gap starting primes:
Gap 148 starts at 2,010,733, gap 150 starts at 13,626,257, difference is 11,615,524.

Earliest difference > 100,000,000 between adjacent prime gap starting primes:
Gap 198 starts at 46,006,769, gap 200 starts at 378,043,979, difference is 332,037,210.

Earliest difference > 1,000,000,000 between adjacent prime gap starting primes:
Gap 276 starts at 649,580,171, gap 278 starts at 4,260,928,601, difference is 3,611,348,430.
</pre>
=={{header|Mathematica}}/{{header|Wolfram Language}}==
<syntaxhighlight lang="mathematica">primes = Prime[Range[10^7]];
gaps = {Differences[primes], Most[primes]} // Transpose;
tmp = GatherBy[gaps, First][[All, 1]];
tmp = SortBy[tmp, First];
starts = Association[Rule @@@ tmp];
set = {Most[tmp[[All, 1]]], Abs@Differences[tmp[[All, 2]]]} // Transpose;
data = Table[{n, k} = SelectFirst[set, Last/*GreaterThan[10^i]];
   {10^i, n, starts[n], n + 2, starts[n + 2], k}, {i, 1, 7}];
StringTemplate["Earliest difference > `` between adjacent prime gap starting primes:
Gap `` starts at ``, gap `` starts at ``, difference is ``."]/*Print @@@ data;</syntaxhighlight>
{{out}}
<pre>Earliest difference > 10 between adjacent prime gap starting primes:
Gap 4 starts at 7, gap 6 starts at 23, difference is 16.
Earliest difference > 100 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1831, difference is 1718.
Earliest difference > 1000 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1831, difference is 1718.
Earliest difference > 10000 between adjacent prime gap starting primes:
Gap 36 starts at 9551, gap 38 starts at 30593, difference is 21042.
Earliest difference > 100000 between adjacent prime gap starting primes:
Gap 70 starts at 173359, gap 72 starts at 31397, difference is 141962.
Earliest difference > 1000000 between adjacent prime gap starting primes:
Gap 100 starts at 396733, gap 102 starts at 1444309, difference is 1047576.
Earliest difference > 10000000 between adjacent prime gap starting primes:
Gap 148 starts at 2010733, gap 150 starts at 13626257, difference is 11615524.</pre>

=={{header|Nim}}==
<syntaxhighlight lang="Nim">import std/[bitops, math, strformat, tables]

type Sieve = object
  data: seq[byte]

proc `[]`(sieve: Sieve; idx: Positive): bool =
  let idx = idx shr 1
  let iByte = idx shr 3
  let iBit = idx and 7
  result = sieve.data[iByte].testBit(iBit)

proc `[]=`(sieve: var Sieve; idx: Positive; val: bool) =
  let idx = idx shr 1
  let iByte = idx shr 3
  let iBit = idx and 7
  if val: sieve.data[iByte].setBit(iBit)
  else: sieve.data[iByte].clearBit(iBit)

proc newSieve(lim: Positive): Sieve =
  result.data = newSeq[byte]((lim + 16) shr 4)

proc initPrimes(lim: Positive): seq[Natural] =
  var composite = newSieve(lim)
  composite[1] = true
  for n in countup(3, sqrt(lim.toFloat).int, 2):
    if not composite[n]:
      for k in countup(n * n, lim, 2 * n):
        composite[k] = true
  for n in countup(3, lim, 2):
    if not composite[n]:
      result.add n

const Limit = 100_000_000
let primes = initPrimes(Limit * 5)
var gapStarts: Table[int, int]
for i in 1..primes.high:
  let gap = primes[i] - primes[i - 1]
  if gap notin gapStarts:
    gapStarts[gap] = primes[i - 1]
var pm = 10
var gap1 = 2
while true:
  while gap1 notin gapStarts:
    inc gap1, 2
  let start1 = gapStarts[gap1]
  let gap2 = gap1 + 2
  if gap2 notin gapStarts:
    gap1 = gap2 + 2
    continue
  let start2 = gapStarts[gap2]
  let diff = abs(start2 - start1)
  if diff > pm:
    echo &"Earliest difference > {pm} between adjacent prime gap starting primes:"
    echo &"Gap {gap1} starts at {start1}, gap {gap2} starts at {start2}, difference is {diff}.\n"
    if pm == Limit: break
    pm *= 10
  else:
    gap1 = gap2
</syntaxhighlight>

{{out}}
<pre>Earliest difference > 10 between adjacent prime gap starting primes:
Gap 4 starts at 7, gap 6 starts at 23, difference is 16.

Earliest difference > 100 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1831, difference is 1718.

Earliest difference > 1000 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1831, difference is 1718.

Earliest difference > 10000 between adjacent prime gap starting primes:
Gap 36 starts at 9551, gap 38 starts at 30593, difference is 21042.

Earliest difference > 100000 between adjacent prime gap starting primes:
Gap 70 starts at 173359, gap 72 starts at 31397, difference is 141962.

Earliest difference > 1000000 between adjacent prime gap starting primes:
Gap 100 starts at 396733, gap 102 starts at 1444309, difference is 1047576.

Earliest difference > 10000000 between adjacent prime gap starting primes:
Gap 148 starts at 2010733, gap 150 starts at 13626257, difference is 11615524.

Earliest difference > 100000000 between adjacent prime gap starting primes:
Gap 198 starts at 46006769, gap 200 starts at 378043979, difference is 332037210.
</pre>

=={{header|Pascal}}==
==={{header|Free Pascal}}===
<syntaxhighlight lang="pascal">program primesieve;
// sieving small ranges of 65536
//{$O+,R+}
{$IFDEF FPC}
  {$MODE DELPHI}{$OPTIMIZATION ON,ALL}{$CODEALIGN proc=32}
  uses
    sysutils;
{$ENDIF}
{$IFDEF WINDOWS}
  {$APPTYPE CONSOLE}
{$ENDIF}

const
  smlPrimes :array [0..10] of Byte = (2,3,5,7,11,13,17,19,23,29,31);
  maxPreSievePrime = 17;
  sieveSize = 1 shl 15;//32768*2 ->max count of FoundPrimes = 6542
type
  tSievePrim = record
                 svdeltaPrime:word;//diff between actual and new prime
                 svSivOfs:word;//-> sieveSize< 1 shl 16
                 svSivNum:LongWord;// 1 shl (16+32) = 2.8e14
               end;
var
{$Align 16}
  //primes up to 1E6-> sieving to 1E12
  sievePrimes : array[0..78497] of tSievePrim;
{$Align 16}
  preSieve :array[0..3*5*7*11*13*17-1] of Byte;
{$Align 16}
  Sieve :array[0..sieveSize-1] of Byte;
{$Align 16}
  FoundPrimes : array[0..6542] of LongWord;
{$Align 16}
  Gaps  : array[0..255] Of Uint64;
{$Align 16}
  Limit,OffSet : Uint64;

  SieveMaxIdx,
  preSieveOffset,
  SieveNum,
  FoundPrimesCnt,
  PrimPos,
  LastInsertedSievePrime :NativeUInt;

procedure CopyPreSieveInSieve;forward;
procedure CollectPrimes;forward;
procedure sieveOneSieve;forward;

procedure preSieveInit;
var
  i,pr,j,umf : NativeInt;
Begin
  fillchar(preSieve[0],SizeOf(preSieve),#1);
  i := 1;// starts with pr = 3
  umf := 1;
  repeat
    IF preSieve[i] <> 0 then
    Begin
      pr := 2*i+1;
      j := i;
      repeat
        preSieve[j] := 0;
        inc(j,pr);
      until j> High(preSieve);
      umf := umf*pr;
    end;
    inc(i);
  until umf>High(preSieve);
  preSieveOffset := 0;
end;

procedure CalcSievePrimOfs(lmt:NativeUint);
var
  i,pr : NativeUInt;
  sq : Uint64;
begin
  pr := 0;
  i := 0;
  repeat
    with sievePrimes[i] do
    Begin
      pr := pr+svdeltaPrime;
      IF sqr(pr)  < (SieveSize*2) then
      Begin
        svSivNum := 0;
        svSivOfs := (pr*pr-1) DIV 2;
      end
      else
      Begin
        SieveMaxIdx := i;
        pr := pr-svdeltaPrime;
        BREAK;
      end;
    end;
    inc(i);
  until i > lmt;

  for i := i to lmt do
  begin
    with sievePrimes[i] do
    Begin
      pr := pr+svdeltaPrime;
      sq := sqr(pr);
      svSivNum := sq DIV (2*SieveSize);
      svSivOfs := ( (sq - Uint64(svSivNum)*(2*SieveSize))-1)DIV 2;
    end;
  end;
end;

procedure InitSieve;
begin
  preSieveOffset := 0;
  SieveNum :=0;
  CalcSievePrimOfs(PrimPos-1);
end;

procedure InsertSievePrimes;
var
  j    :NativeUINt;
  i,pr : NativeUInt;
begin
  i := 0;
  //ignore first primes already sieved with
  if SieveNum = 0 then
    repeat
      inc(i);
    until FoundPrimes[i] > maxPreSievePrime;

  pr :=0;
  j := Uint64(SieveNum)*SieveSize*2-LastInsertedSievePrime;
  with sievePrimes[PrimPos] do
  Begin
    pr := FoundPrimes[i];
    svdeltaPrime := pr+j;
    j := pr;
    end;
  inc(PrimPos);
  for i := i+1 to FoundPrimesCnt-1 do
  Begin
    IF PrimPos > High(sievePrimes) then
      BREAK;
    with sievePrimes[PrimPos] do
    Begin
      pr := FoundPrimes[i];
      svdeltaPrime := (pr-j);
      j := pr;
    end;
    inc(PrimPos);
  end;
  LastInsertedSievePrime :=Uint64(SieveNum)*(SieveSize*2)+pr;
end;

procedure sievePrimesInit;
var
  i,j,pr:NativeInt;
Begin
  LastInsertedSievePrime := 0;

  PrimPos := 0;
  preSieveOffset := 0;
  SieveNum :=0;
  CopyPreSieveInSieve;
  i := 1; // start with 3
  repeat
    while Sieve[i] = 0 do
      inc(i);
    pr := 2*i+1;
    inc(i);
    j := ((pr*pr)-1) DIV 2;
    if j > High(Sieve) then
      BREAK;
    repeat
      Sieve[j] := 0;
      inc(j,pr);
    until j > High(Sieve);
  until false;

  CollectPrimes;
  InsertSievePrimes;
  IF PrimPos < High(sievePrimes) then
  Begin
    InitSieve;
    //Erste Sieb nochmals, aber ohne Eintrag
    CopyPreSieveInSieve;
    sieveOneSieve;
    repeat
      inc(SieveNum);
      CopyPreSieveInSieve;
      sieveOneSieve;
      CollectPrimes;
      InsertSievePrimes;
   until PrimPos > High(sievePrimes);
  end;
end;

procedure OutGaps(g1,g2,delta:NativeUint);
begin
  if g2= 0  then
    writeln(2*g1:4,2*g1+2:4,delta:13,Gaps[g1]:13,Gaps[g1+1]:13)
  else
   writeln(2*g2-1:4,2*g1:4,delta:13,Gaps[g1-1]:13,Gaps[g1]:13);
end;
function GetDiffval(val1,val2: NativeInt): NativeInt;
begin
  if val1*val2 = 0 then
    EXIT(0);
  dec(val1,val2);
  if val1<0 then
    val1 :=-val1;
  GetDiffval := val1;
end;

procedure CheckGaps;
var
  val1,val2,val3,i,DekaLimit : NativeInt;
Begin
  writeln('Gap1 Gap2   difference       first       second  prime');
  dekaLimit := 10;
  i := 1;
  repeat
    val1 := Gaps[i];
    if val1 <> 0 then
    Begin
      val2 := GetDiffval(val1,Gaps[i-1]);
      val3 := GetDiffval(val1,Gaps[i+1]);
      while (val2>DekaLimit) or (val3>DekaLimit) do
      begin
        writeln(DekaLimit:21,'<');
        if val2 = 0 then
        begin
          if val3 > 0 then
            OutGaps(i,0,val3);
        end
        else
        begin
          if val3 = 0 then
            OutGaps(i,1,val2)
          else
          Begin
            if val3 > val2 then
              OutGaps(i,0,val3)
            else
              OutGaps(i,1,val2);
          end;
        end;
        DekaLimit := 10*DekaLimit;
      end;
    end;
    inc(i);
  until i>=254;
end;

procedure CopyPreSieveInSieve;
var
  lmt : NativeInt;
Begin
  lmt := preSieveOffset+sieveSize;
  lmt := lmt-(High(preSieve)+1);
  IF lmt<= 0 then
  begin
    Move(preSieve[preSieveOffset],Sieve[0],sieveSize);
    if lmt <> 0 then
      inc(preSieveOffset,sieveSize)
    else
      preSieveOffset := 0;
  end
  else
  begin
    Move(preSieve[preSieveOffset],Sieve[0],sieveSize-lmt);
    Move(preSieve[0],Sieve[sieveSize-lmt],lmt);
    preSieveOffset := lmt
  end;
end;

procedure CollectPrimes;
var
   pSieve : pbyte;
   pFound : pLongWord;
   i,idx : NativeInt;
Begin
  pFound := @FoundPrimes[0];
  idx := 0;
  i := 0;
  IF SieveNum = 0 then
  Begin
    repeat
      pFound[idx] := smlPrimes[idx];
      inc(idx);
    until smlPrimes[idx]>maxPreSievePrime;
    i := (smlPrimes[idx] -1) DIV 2;
  end;

  pSieve := @Sieve[0];
  repeat
    pFound[idx]:= 2*i+1;
    inc(idx,pSieve[i]);
    inc(i);
  until i>High(Sieve);
  FoundPrimesCnt:= idx;
end;

procedure sieveOneSieve;
var
  i,j,pr,dSievNum :NativeUint;
Begin
  pr := 0;
  For i := 0 to SieveMaxIdx do
    with sievePrimes[i] do
    begin
      pr := pr+svdeltaPrime;
      IF svSivNum = sieveNum then
      Begin
        j := svSivOfs;
        repeat
          Sieve[j] := 0;
          inc(j,pr);
        until j > High(Sieve);

        dSievNum := j DIV SieveSize;
        svSivOfs := j-dSievNum*SieveSize;
        inc(svSivNum,dSievNum);
      end;
    end;

  i := SieveMaxIdx+1;
  repeat
    if i > High(SievePrimes) then
      BREAK;
    with sievePrimes[i] do
    begin
      if svSivNum > sieveNum then
      Begin
        SieveMaxIdx := I-1;
        Break;
      end;
      pr := pr+svdeltaPrime;
      j := svSivOfs;
      repeat
        Sieve[j] := 0;
        inc(j,pr);
      until j > High(Sieve);
      dSievNum := j DIV SieveSize;
      svSivOfs := j-dSievNum*SieveSize;
      inc(svSivNum,dSievNum);
      inc(i);
    end;
  until false;
end;

var
  T1,T0,CNT,ActPrime,LastPrime,delta : Int64;
  i: Int32;

begin
  T0 := GetTickCount64;
  Limit := 10*1000*1000*1000;//158*1000*1000*1000;
  preSieveInit;
  sievePrimesInit;

  InitSieve;
  offset := 0;
  Cnt := 1;//==2
  LastPrime := 2;


  repeat
    CopyPreSieveInSieve;sieveOneSieve;CollectPrimes;
    inc(Cnt,FoundPrimesCnt);
    //collect first occurrence of gap
    i := 0;
    repeat
      ActPrime := Offset+FoundPrimes[i];
      delta := (ActPrime - LastPrime) shr 1;
      If Gaps[delta] = 0 then
        Gaps[delta] := LastPrime;
      LastPrime := ActPrime;
      inc(i);
    until (i >= FoundPrimesCnt);

    inc(SieveNum);
    inc(offset,2*SieveSize);
  until SieveNum > (Limit DIV (2*SieveSize));
  CheckGaps;
  T1 := GetTickCount64;
  OffSet := Uint64(SieveNum-1)*(2*SieveSize);


  i := FoundPrimesCnt;
  repeat
    dec(i);
    dec(cnt);
  until (i = 0) OR (OffSet+FoundPrimes[i]<Limit);
  writeln;
  writeln(cnt,' in ',Limit,' takes ',T1-T0,' ms');
  {$IFDEF WINDOWS}
  writeln('Press <Enter>');readln;
  {$ENDIF}
end.</syntaxhighlight>
{{out|@TIO.RUN}}
<pre>
Gap1 Gap2   difference       first       second  prime
                   10<
   4   6           16            7           23
                  100<
  14  16         1718          113         1831
                 1000<
  14  16         1718          113         1831
                10000<
  36  38        21042         9551        30593
               100000<
  70  72       141962       173359        31397
              1000000<
 100 102      1047576       396733      1444309
             10000000<
 148 150     11615524      2010733     13626257
            100000000<
 198 200    332037210     46006769    378043979

50847534 in 1000000000 takes 886 ms
//@home primes til 158,000,000,000 like C++
 276 278   3611348430    649580171   4260928601
          10000000000<
 332 334  24933958388   5893180121  30827138509
         100000000000<
 386 388 121560146636  35238645587 156798792223

6385991032 in 158000000000 takes 247532 ms
</pre>

=={{header|Perl}}==
<syntaxhighlight lang="perl">#!/usr/bin/perl

use strict; # https://rosettacode.org/wiki/Earliest_difference_between_prime_gaps
use warnings;
use ntheory qw( primes );

my @gaps;
my $primeref = primes( 1e9 );
for my $i ( 2 .. $#$primeref )
  {
  my $diff = $primeref->[$i] - $primeref->[$i - 1];
  $gaps[ $diff >> 1 ] //= $primeref->[$i - 1];
  }
my %first;
for my $i ( 1 .. $#gaps )
  {
  defined $gaps[$i] && defined $gaps[$i-1] or next;
  my $diff = abs $gaps[$i] - $gaps[$i-1];
  for my $m ( map 10 ** $_, 1 .. 10 )
    {
    $diff > $m and !$first{$m}++ and
      print "above $m gap @{[$i * 2 - 2 ]} abs( $gaps[$i-1] - $gaps[$i] )\n";
    }
  }</syntaxhighlight>
{{out}}
<pre>
above 10 gap 4 abs( 7 - 23 )
above 100 gap 14 abs( 113 - 1831 )
above 1000 gap 14 abs( 113 - 1831 )
above 10000 gap 36 abs( 9551 - 30593 )
above 100000 gap 70 abs( 173359 - 31397 )
above 1000000 gap 100 abs( 396733 - 1444309 )
above 10000000 gap 148 abs( 2010733 - 13626257 )
above 100000000 gap 198 abs( 46006769 - 378043979 )
</pre>

=={{header|Phix}}==
{{trans|Wren}}
<!--<syntaxhighlight lang="phix">(phixonline)-->
 <span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
 <span style="color: #008080;">constant</span> <span style="color: #000000;">limit</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">platform</span><span style="color: #0000FF;">()=</span><span style="color: #004600;">JS</span><span style="color: #0000FF;">?</span><span style="color: #000000;">1e7</span><span style="color: #0000FF;">:</span><span style="color: #000000;">1e8</span><span style="color: #0000FF;">),</span>
          <span style="color: #000000;">gslim</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">250</span>
 <span style="color: #004080;">sequence</span> <span style="color: #000000;">primes</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">get_primes_le</span><span style="color: #0000FF;">(</span><span style="color: #000000;">limit</span><span style="color: #0000FF;">*</span><span style="color: #000000;">4</span><span style="color: #0000FF;">),</span>
          <span style="color: #000000;">gapstarts</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">gslim</span><span style="color: #0000FF;">)</span>
 <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">primes</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">gap</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">primes</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]-</span><span style="color: #000000;">primes</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
     <span style="color: #008080;">if</span> <span style="color: #000000;">gapstarts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">gap</span><span style="color: #0000FF;">]=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
         <span style="color: #000000;">gapstarts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">gap</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">primes</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> 
 <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
 
 <span style="color: #004080;">integer</span> <span style="color: #000000;">pm</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">10</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">gap1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span>
 <span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
     <span style="color: #008080;">while</span> <span style="color: #000000;">gapstarts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">gap1</span><span style="color: #0000FF;">]=</span><span style="color: #000000;">0</span> <span style="color: #008080;">do</span> <span style="color: #000000;">gap1</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">2</span> <span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">start1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">gapstarts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">gap1</span><span style="color: #0000FF;">],</span>
             <span style="color: #000000;">gap2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">gap1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">2</span>
     <span style="color: #008080;">if</span> <span style="color: #000000;">gapstarts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">gap2</span><span style="color: #0000FF;">]=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
         <span style="color: #000000;">gap1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">gap2</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">2</span>
     <span style="color: #008080;">else</span>
         <span style="color: #004080;">integer</span> <span style="color: #000000;">start2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">gapstarts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">gap2</span><span style="color: #0000FF;">],</span>
                 <span style="color: #000000;">diff</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">start2</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">start1</span><span style="color: #0000FF;">)</span>
         <span style="color: #008080;">if</span> <span style="color: #000000;">diff</span><span style="color: #0000FF;">></span><span style="color: #000000;">pm</span> <span style="color: #008080;">then</span>
             <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Earliest difference &gt;%,d between adjacent prime gap starting primes:\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">pm</span><span style="color: #0000FF;">})</span>
             <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Gap %d starts at %,d, gap %d starts at %,d, difference is %,d.\n\n"</span><span style="color: #0000FF;">,</span>
                          <span style="color: #0000FF;">{</span><span style="color: #000000;">gap1</span><span style="color: #0000FF;">,</span>        <span style="color: #000000;">start1</span><span style="color: #0000FF;">,</span>  <span style="color: #000000;">gap2</span><span style="color: #0000FF;">,</span>        <span style="color: #000000;">start2</span><span style="color: #0000FF;">,</span>            <span style="color: #000000;">diff</span><span style="color: #0000FF;">})</span>
             <span style="color: #008080;">if</span> <span style="color: #000000;">pm</span><span style="color: #0000FF;">=</span><span style="color: #000000;">limit</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
             <span style="color: #000000;">pm</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">10</span>
         <span style="color: #008080;">else</span>
             <span style="color: #000000;">gap1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">gap2</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<!--</syntaxhighlight>-->
Output same as Wren. Takes 5s on desktop/Phix but would take 17s under p2js so limited that to keep it under 2s.
A limit of 1e9 needs 64 bit (hence not p2js compatible), and a multiplier of 5, and a gslim of 354, and takes 2 mins 43s, and nearly killed my poor little box, but matched the output of Julia, which managed it in 40s (and with no signs of any stress). Python needed more memory than I've got for 1e9, but took 15s for a limit of 1e8, while Wren (bless, also 1e8) took just over 3 minutes (an old i5/8GB/w10).

=={{header|Python}}==
{{trans|Julia, Wren}}
<syntaxhighlight lang="python">""" https://rosettacode.org/wiki/Earliest_difference_between_prime_gaps """

from primesieve import primes

LIMIT = 10**9
pri = primes(LIMIT * 5)
gapstarts = {}
for i in range(1, len(pri)):
    if pri[i] - pri[i - 1] not in gapstarts:
        gapstarts[pri[i] - pri[i - 1]] = pri[i - 1]

PM, GAP1, = 10, 2
while True:
    while GAP1 not in gapstarts:
        GAP1 += 2
    start1 = gapstarts[GAP1]
    GAP2 = GAP1 + 2
    if GAP2 not in gapstarts:
        GAP1 = GAP2 + 2
        continue
    start2 = gapstarts[GAP2]
    diff = abs(start2 - start1)
    if diff > PM:
        print(f"Earliest difference >{PM: ,} between adjacent prime gap starting primes:")
        print(f"Gap {GAP1} starts at{start1: ,}, gap {GAP2} starts at{start2: ,}, difference is{diff: ,}.\n")
        if PM == LIMIT:
            break
        PM *= 10
    else:
        GAP1 = GAP2
</syntaxhighlight>{{out}}Same as Raku, Wren and Julia versions.

=={{header|Raku}}==

1e1 through 1e7 are pretty speedy (less than 5 seconds total). 1e8 and 1e9 take several minutes.
<syntaxhighlight lang="raku" line>use Math::Primesieve;
use Lingua::EN::Numbers;

my $iterator = Math::Primesieve::iterator.new;
my @gaps;
my $last = 2;

for 1..9 {
    my $m = exp $_, 10;
    my $this;
    loop {
        $this = (my $p = $iterator.next) - $last;
        if !@gaps[$this].defined {
             @gaps[$this]= $last;
             check-gap($m, $this, @gaps) && last
               if $this > 2 and @gaps[$this - 2].defined and (abs($last - @gaps[$this - 2]) > $m);
        }
        $last = $p;
    }
}

sub check-gap ($n, $this, @p) {
    my %upto = @p[^$this].pairs.grep: *.value;
    my @upto = (1..$this).map: { last unless %upto{$_ * 2}; %upto{$_ * 2} }
    my $key = @upto.rotor(2=>-1).first( {.sink; abs(.[0] - .[1]) > $n}, :k );
    return False unless $key;
    say "Earliest difference > {comma $n} between adjacent prime gap starting primes:";
    printf "Gap %s starts at %s, gap %s starts at %s, difference is %s\n\n",
    |(2 * $key + 2, @upto[$key], 2 * $key + 4, @upto[$key+1], abs(@upto[$key] - @upto[$key+1]))».&comma;
    True
}</syntaxhighlight>
{{out}}
<pre>Earliest difference > 10 between adjacent prime gap starting primes:
Gap 4 starts at 7, gap 6 starts at 23, difference is 16

Earliest difference > 100 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1,831, difference is 1,718

Earliest difference > 1,000 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1,831, difference is 1,718

Earliest difference > 10,000 between adjacent prime gap starting primes:
Gap 36 starts at 9,551, gap 38 starts at 30,593, difference is 21,042

Earliest difference > 100,000 between adjacent prime gap starting primes:
Gap 70 starts at 173,359, gap 72 starts at 31,397, difference is 141,962

Earliest difference > 1,000,000 between adjacent prime gap starting primes:
Gap 100 starts at 396,733, gap 102 starts at 1,444,309, difference is 1,047,576

Earliest difference > 10,000,000 between adjacent prime gap starting primes:
Gap 148 starts at 2,010,733, gap 150 starts at 13,626,257, difference is 11,615,524

Earliest difference > 100,000,000 between adjacent prime gap starting primes:
Gap 198 starts at 46,006,769, gap 200 starts at 378,043,979, difference is 332,037,210

Earliest difference > 1,000,000,000 between adjacent prime gap starting primes:
Gap 276 starts at 649,580,171, gap 278 starts at 4,260,928,601, difference is 3,611,348,430</pre>

=={{header|Rust}}==
<syntaxhighlight lang="rust">// [dependencies]
// primal = "0.3"

fn main() {
    use std::collections::HashMap;

    let mut primes = primal::Primes::all();
    let mut last_prime = primes.next().unwrap();
    let mut gap_starts = HashMap::new();

    let mut find_gap_start = move |gap: usize| -> usize {
        if let Some(start) = gap_starts.get(&gap) {
            return *start;
        }
        loop {
            let prev = last_prime;
            last_prime = primes.next().unwrap();
            let diff = last_prime - prev;
            if !gap_starts.contains_key(&diff) {
                gap_starts.insert(diff, prev);
            }
            if gap == diff {
                return prev;
            }
        }
    };

    let limit = 100000000000;

    let mut pm = 10;
    let mut gap1 = 2;
    loop {
        let start1 = find_gap_start(gap1);
        let gap2 = gap1 + 2;
        let start2 = find_gap_start(gap2);
        let diff = if start2 > start1 {
            start2 - start1
        } else {
            start1 - start2
        };
        if diff > pm {
            println!(
                "Earliest difference > {} between adjacent prime gap starting primes:\n\
                Gap {} starts at {}, gap {} starts at {}, difference is {}.\n",
                pm, gap1, start1, gap2, start2, diff
            );
            if pm == limit {
                break;
            }
            pm *= 10;
        } else {
            gap1 = gap2;
        }
    }
}</syntaxhighlight>

{{out}}
<pre>
Earliest difference > 10 between adjacent prime gap starting primes:
Gap 4 starts at 7, gap 6 starts at 23, difference is 16.

Earliest difference > 100 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1831, difference is 1718.

Earliest difference > 1000 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1831, difference is 1718.

Earliest difference > 10000 between adjacent prime gap starting primes:
Gap 36 starts at 9551, gap 38 starts at 30593, difference is 21042.

Earliest difference > 100000 between adjacent prime gap starting primes:
Gap 70 starts at 173359, gap 72 starts at 31397, difference is 141962.

Earliest difference > 1000000 between adjacent prime gap starting primes:
Gap 100 starts at 396733, gap 102 starts at 1444309, difference is 1047576.

Earliest difference > 10000000 between adjacent prime gap starting primes:
Gap 148 starts at 2010733, gap 150 starts at 13626257, difference is 11615524.

Earliest difference > 100000000 between adjacent prime gap starting primes:
Gap 198 starts at 46006769, gap 200 starts at 378043979, difference is 332037210.

Earliest difference > 1000000000 between adjacent prime gap starting primes:
Gap 276 starts at 649580171, gap 278 starts at 4260928601, difference is 3611348430.

Earliest difference > 10000000000 between adjacent prime gap starting primes:
Gap 332 starts at 5893180121, gap 334 starts at 30827138509, difference is 24933958388.

Earliest difference > 100000000000 between adjacent prime gap starting primes:
Gap 386 starts at 35238645587, gap 388 starts at 156798792223, difference is 121560146636.

</pre>

=={{header|Sidef}}==
<syntaxhighlight lang="ruby">func prime_gap_records(upto) {

    var gaps = []
    var p = 3

    each_prime(p.next_prime, upto, {|q|
        gaps[q-p] := p
        p = q
    })

    gaps.grep { defined(_) }
}

var gaps = prime_gap_records(1e8)

for m in (1 .. gaps.max.len) {
    gaps.each_cons(2, {|p,q|
        if (abs(q-p) > 10**m) {
            say "10^#{m} -> (#{p}, #{q}) with gaps (#{
            p.next_prime-p}, #{q.next_prime-q}) and difference #{abs(q-p)}"
            break
        }
    })
}</syntaxhighlight>
{{out}}
<pre>
10^1 -> (7, 23) with gaps (4, 6) and difference 16
10^2 -> (113, 1831) with gaps (14, 16) and difference 1718
10^3 -> (113, 1831) with gaps (14, 16) and difference 1718
10^4 -> (9551, 30593) with gaps (36, 38) and difference 21042
10^5 -> (173359, 31397) with gaps (70, 72) and difference 141962
10^6 -> (396733, 1444309) with gaps (100, 102) and difference 1047576
10^7 -> (2010733, 13626257) with gaps (148, 150) and difference 11615524
</pre>

=={{header|Wren}}==
{{libheader|Wren-math}}
{{libheader|Wren-fmt}}
This uses a segmented sieve to avoid running out of memory when looking for the earliest difference above 1 billion. Takes a little over 5½ minutes to run (25 seconds to reach first 100 million) on my machine (core i7, 32GB RAM, Ubuntu 20.04).
<syntaxhighlight lang="ecmascript">import "./math" for Int
import "/fmt" for Fmt
 
var limit = 1e9
var gapStarts = {}
var primes = Int.segmentedSieve(limit * 5, 8 * 1024 * 1024) // 8 MB cache
for (i in 1...primes.count) {
    var gap = primes[i] - primes[i-1]
    if (!gapStarts[gap]) gapStarts[gap] = primes[i-1]
}
var pm = 10
var gap1 = 2
while (true) {
    while (!gapStarts[gap1]) gap1 = gap1 + 2
    var start1 = gapStarts[gap1]
    var gap2 = gap1 + 2
    if (!gapStarts[gap2]) {
        gap1 = gap2 + 2
        continue
    }
    var start2 = gapStarts[gap2]
    var diff = (start2 - start1).abs
    if (diff > pm) {
        Fmt.print("Earliest difference > $,d between adjacent prime gap starting primes:", pm)
        Fmt.print("Gap $d starts at $,d, gap $d starts at $,d, difference is $,d.\n", gap1, start1, gap2, start2, diff)
        if (pm == limit) break
        pm = pm * 10
    } else {
        gap1 = gap2
    }
}</syntaxhighlight>

{{out}}
<pre>
Earliest difference > 10 between adjacent prime gap starting primes:
Gap 4 starts at 7, gap 6 starts at 23, difference is 16.

Earliest difference > 100 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1,831, difference is 1,718.

Earliest difference > 1,000 between adjacent prime gap starting primes:
Gap 14 starts at 113, gap 16 starts at 1,831, difference is 1,718.

Earliest difference > 10,000 between adjacent prime gap starting primes:
Gap 36 starts at 9,551, gap 38 starts at 30,593, difference is 21,042.

Earliest difference > 100,000 between adjacent prime gap starting primes:
Gap 70 starts at 173,359, gap 72 starts at 31,397, difference is 141,962.

Earliest difference > 1,000,000 between adjacent prime gap starting primes:
Gap 100 starts at 396,733, gap 102 starts at 1,444,309, difference is 1,047,576.

Earliest difference > 10,000,000 between adjacent prime gap starting primes:
Gap 148 starts at 2,010,733, gap 150 starts at 13,626,257, difference is 11,615,524.

Earliest difference > 100,000,000 between adjacent prime gap starting primes:
Gap 198 starts at 46,006,769, gap 200 starts at 378,043,979, difference is 332,037,210.

Earliest difference > 1,000,000,000 between adjacent prime gap starting primes:
Gap 276 starts at 649,580,171, gap 278 starts at 4,260,928,601, difference is 3,611,348,430.
</pre>