# Earliest difference between prime gaps

**Earliest difference between prime gaps**

You are encouraged to solve this task according to the task description, using any language you may know.

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.

gap | minimal starting prime |
ending 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 |

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.

## ALGOL 68[edit]

```
BEGIN # find the differences between primes for each value of the gap #
# the prime and the next #
PR read "primes.incl.a68" PR # include prime utilities #
INT max prime = 2 000 000; # maximum prime we will consider #
[]BOOL prime = PRIMESIEVE max prime; # sieve the primes to max prime #
[ 1 : max prime OVER 2 ]INT start prime; # index of the first prime with #
# gap of subscript / 2 #
# find the prime gaps #
FOR i TO UPB start prime DO start prime[ i ] := 0 OD;
INT prev prime := 3;
FOR i FROM 5 BY 2 TO UPB prime DO
IF prime[ i ] THEN
INT gap = ( i - prev prime ) OVER 2;
IF start prime[ gap ] = 0 THEN
start prime[ gap ] := prev prime
FI;
prev prime := i
FI
OD;
# to reiterate the task: we must find the earliest start primes where #
# the distance betweeen the gaps is 10, 100, 1000, 10 000 etc. #
# The distance is the distance between the start prime with gap g and #
# start prime with the a gap of g + 2, e.g. 3 has a gap of 2 as the next #
# prime is 5, 7 has a gap of 4 as the next prime is 11, so the distance #
# is: 7 - 3 = 4 #
# shows a prime gap #
PROC show gap = ( INT start pos )VOID:
print( ( whole( start prime[ start pos ], 0 )
, "(", whole( start pos * 2, 0 ),")"
, whole( start prime[ start pos ] + ( start pos * 2 ), 0 )
)
);
# shows a prime gap distance #
PROC show distance = ( INT gap, pos )VOID:
BEGIN
print( ( "First distance > ", whole( gap, 0 )
, " betweeen prime gaps:", newline
, " ", whole( ABS ( start prime[ pos + 1 ] - start prime[ pos ] ), 0 )
, " between "
)
);
show gap( pos );
print( " and " );
show gap( pos + 1 );
print( ( newline ) )
END # show distance # ;
INT g10 := 0, g100 := 0, gt := 0, g10t := 0, g100t := 0, gm := 0;
FOR i TO UPB start prime - 1 DO
IF start prime[ i ] /= 0 AND start prime[ i + 1 ] /= 0 THEN
INT distance = ABS ( start prime[ i + 1 ] - start prime[ i ] );
IF distance > 10
THEN
IF g10 = 0 THEN g10 := i FI
FI;
IF distance > 100
THEN
IF g100 = 0 THEN g100 := i FI
FI;
IF distance > 1 000
THEN
IF gt = 0 THEN gt := i FI
FI;
IF distance > 10 000
THEN
IF g10t = 0 THEN g10t := i FI
FI;
IF distance > 100 000
THEN
IF g100t = 0 THEN g100t := i FI
FI;
IF distance > 1 000 000
THEN
IF gm = 0 THEN gm := i FI
FI
FI
OD;
show distance( 10, g10 );
show distance( 100, g100 );
show distance( 1 000, gt );
show distance( 10 000, g10t );
show distance( 100 000, g100t );
show distance( 1 000 000, gm )
END
```

- Output:

First distance > 10 betweeen prime gaps: 16 between 7(4)11 and 23(6)29 First distance > 100 betweeen prime gaps: 1718 between 113(14)127 and 1831(16)1847 First distance > 1000 betweeen prime gaps: 1718 between 113(14)127 and 1831(16)1847 First distance > 10000 betweeen prime gaps: 21042 between 9551(36)9587 and 30593(38)30631 First distance > 100000 betweeen prime gaps: 141962 between 173359(70)173429 and 31397(72)31469 First distance > 1000000 betweeen prime gaps: 1047576 between 396733(100)396833 and 1444309(102)1444411

## C++[edit]

```
#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;
}
}
}
```

- Output:

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.

## F#[edit]

This task uses Extensible Prime Generator (F#)

```
// 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))}")
```

- Output:

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

## Go[edit]

```
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
}
}
}
```

- Output:

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.

## J[edit]

Implementation:

```
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
}}
```

Task examples:

```
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
```

## Java[edit]

Uses the PrimeGenerator class from Extensible prime generator#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;
}
}
}
```

- Output:

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.

## Julia[edit]

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.

```
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()
```

- Output:

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.

## Mathematica/Wolfram Language[edit]

```
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;
```

- Output:

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.

## Nim[edit]

To find the primes, we used a sieve of Erathostenes for odd numbers only where each element is a bit.

```
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
```

- Output:

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.

## Pascal[edit]

### Free Pascal[edit]

```
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.
```

- @TIO.RUN:

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

## Perl[edit]

```
#!/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";
}
}
```

- Output:

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 )

## Phix[edit]

with javascript_semantics constant limit = iff(platform()=JS?1e7:1e8), gslim = 250 sequence primes = get_primes_le(limit*4), gapstarts = repeat(0,gslim) for i=2 to length(primes) do integer gap = primes[i]-primes[i-1] if gapstarts[gap]=0 then gapstarts[gap] = primes[i-1] end if end for integer pm = 10, gap1 = 2 while true do while gapstarts[gap1]=0 do gap1 += 2 end while integer start1 = gapstarts[gap1], gap2 = gap1 + 2 if gapstarts[gap2]=0 then gap1 = gap2 + 2 else integer start2 = gapstarts[gap2], diff = abs(start2 - start1) if diff>pm then printf(1,"Earliest difference >%,d between adjacent prime gap starting primes:\n",{pm}) printf(1,"Gap %d starts at %,d, gap %d starts at %,d, difference is %,d.\n\n", {gap1, start1, gap2, start2, diff}) if pm=limit then exit end if pm *= 10 else gap1 = gap2 end if end if end while

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).

## Python[edit]

```
""" 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
```

- Output:

## Raku[edit]

1e1 through 1e7 are pretty speedy (less than 5 seconds total). 1e8 and 1e9 take several minutes.

```
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]))».,
True
}
```

- Output:

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

## Rust[edit]

```
// [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;
}
}
}
```

- Output:

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.

## Sidef[edit]

```
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
}
})
}
```

- Output:

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

## Wren[edit]

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).

```
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
}
}
```

- Output:

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.