Largest difference between adjacent primes
- Task
Find and show on this page the largest difference between adjacent primes under 1,000,000.
ALGOL 68
Note, to run this with Algol 68G, the command line option --heap 256M should be specified to allow a larger heap size.
As with the Wren, Phix, etc. samples, shows the gaps at a few other places.
<lang algol68>BEGIN # find the largest gap between adjacent primes up to 10 000 000 #
PR read "primes.incl.a68" PR
[]BOOL prime = PRIMESIEVE 10 000 000; # sieve the primes to 10 000 000 #
# find the largest gap between primes #
INT max gap := 0; # gap between 2 and 3 #
INT max prime := 0; # the prime with the maximum gap #
INT max prev := 0; # the prime before the maximum gap #
INT prev prime := 2; # the previous prime #
INT power of 10 := 10; # next number to print the gap for #
FOR i FROM 3 BY 2 TO UPB prime DO
IF prime[ i ] THEN
# have a prime #
INT gap = i - prev prime;
IF gap > max gap THEN
# have a bigger gap than before #
max prime := i;
max prev := prev prime;
max gap := gap
FI;
prev prime := i
FI;
IF i = power of 10 - 1 THEN
# have reached the next power of 10 - report the max gap so far #
print( ( "Largest gap between primes up to ", whole( power of 10, -9 )
, ": ", whole( max gap, -9 ), " between "
, whole( max prev, 0 ), " and ", whole( max prime, 0 )
, newline
)
);
power of 10 *:= 10
FI
OD
END</lang>
- Output:
Largest gap between primes up to 10: 2 between 3 and 5 Largest gap between primes up to 100: 8 between 89 and 97 Largest gap between primes up to 1000: 20 between 887 and 907 Largest gap between primes up to 10000: 36 between 9551 and 9587 Largest gap between primes up to 100000: 72 between 31397 and 31469 Largest gap between primes up to 1000000: 114 between 492113 and 492227 Largest gap between primes up to 10000000: 154 between 4652353 and 4652507
C
<lang c>#include<stdio.h>
- include<stdlib.h>
int isprime( int p ) {
int i;
if(p==2) return 1;
if(!(p%2)) return 0;
for(i=3; i*i<=p; i+=2) {
if(!(p%i)) return 0;
}
return 1;
}
int nextprime( int p ) {
int i=0; if(p==0) return 2; if(p<3) return p+1; while(!isprime(++i + p)); return i+p;
}
int main(void) {
int i=3, j=5, champ=3, champj=5, record=2;
for(i=3;j<=1000000;i=j) {
j=nextprime(i);
if(j-i>record) {
champ=i;
champj=j;
record = j-i;
}
}
printf( "The largest difference was %d, between %d and %d.\n", record, champ, champj );
return 0;
}</lang>
- Output:
The largest difference was 114, between 492113 and 492227.
C++
<lang cpp>#include <iostream>
- include <locale>
- include <primesieve.hpp>
int main() {
std::cout.imbue(std::locale(""));
const uint64_t limit = 10000000000;
uint64_t max_diff = 0;
primesieve::iterator pi;
uint64_t p1 = pi.next_prime();
for (uint64_t p = 10;;) {
uint64_t p2 = pi.next_prime();
if (p2 >= p) {
std::cout << "Largest gap between primes under " << p << " is "
<< max_diff << ".\n";
if (p == limit)
break;
p *= 10;
}
if (p2 - p1 > max_diff)
max_diff = p2 - p1;
p1 = p2;
}
}</lang>
- Output:
Largest gap between primes under 10 is 2. Largest gap between primes under 100 is 8. Largest gap between primes under 1,000 is 20. Largest gap between primes under 10,000 is 36. Largest gap between primes under 100,000 is 72. Largest gap between primes under 1,000,000 is 114. Largest gap between primes under 10,000,000 is 154. Largest gap between primes under 100,000,000 is 220. Largest gap between primes under 1,000,000,000 is 282. Largest gap between primes under 10,000,000,000 is 354.
F#
This task uses Extensible Prime Generator (F#) <lang fsharp> // Largest difference between adjacent primes. Nigel Galloway: November 22nd., 2021 let n,g=primes32()|>Seq.takeWhile((>)1000000)|>Seq.pairwise|>Seq.maxBy(fun(n,g)->g-n) in printfn $"%d{g}-%d{n}=%d{g-n}" </lang>
- Output:
492227-492113=114
Factor
See Largest difference between adjacent primes/Factor for a detailed explanation because why not?
<lang factor>USING: arrays formatting kernel lists lists.lazy math math.order math.primes.lists sequences ;
lprimes dup cdr lzip [ first2 2dup swap - -rot 3array ] lmap-lazy [ second 1e6 < ] lwhile { 0 } [ max ] foldl
"Largest difference in adjacent primes under a million: %d between %d and %d.\n" vprintf</lang>
- Output:
Largest difference in adjacent primes under a million: 114 between 492113 and 492227.
FreeBASIC
<lang freebasic>#include "isprime.bas"
function nextprime( n as uinteger ) as uinteger
'finds the next prime after n
if n = 0 then return 2
if n < 3 then return n + 1
dim as integer q = n + 2
while not isprime(q)
q+=2
wend
return q
end function
dim as uinteger i = 5, j, champ=3, champj=5, record=2 while i<1000000
j = nextprime(i)
if j-i > record then
champ = i
champj = j
record = j - i
end if
i = j
wend
print using "The largest difference was ####, between ####### and #######";record;champ;champj</lang>
- Output:
The largest difference was 114 between 492113 and 492227
GW-BASIC
<lang gwbasic>10 R=2 : P=3 : P2=0 20 GOSUB 190 30 IF P2>1000000! THEN GOTO 70 40 IF P2-P > R THEN GOSUB 270 50 P=P2 60 GOTO 20 70 PRINT "The biggest difference was ";R;", between ";C1;" and ";C2 80 END 90 REM tests if a number is prime 100 Q=0 110 IF P = 2 THEN Q = 1:RETURN 120 IF P=3 THEN Q=1:RETURN 130 I=1 140 I=I+1 150 IF INT(P/I)*I = P THEN RETURN 160 IF I*I<=P THEN GOTO 140 170 Q = 1 180 RETURN 190 REM finds the next prime after P, result in P2 200 IF P = 0 THEN P2 = 2: RETURN 210 IF P<3 THEN P2 = P + 1: RETURN 220 T = P 230 P = P + 1 240 GOSUB 90 250 IF Q = 1 THEN P2 = P: P = T: RETURN 260 GOTO 230 270 C1=P 280 C2 = P2 290 R = P2 - P 300 RETURN</lang>
Julia
<lang julia>using Primes
function maxprimeinterval(nmax)
pri = primes(nmax)
diffs = [pri[i] - pri[i - 1] for i in 2:length(pri)]
diff, idx = findmax(diffs)
println("The maximum prime interval in primes up to $nmax is $diff: for example at [$(pri[idx]), $(pri[idx + 1])].")
end
foreach(n -> maxprimeinterval(10^n), 1:10)
</lang>
- Output:
The maximum prime interval in primes up to 10 is 2: for example at [3, 5]. The maximum prime interval in primes up to 100 is 8: for example at [89, 97]. The maximum prime interval in primes up to 1000 is 20: for example at [887, 907]. The maximum prime interval in primes up to 10000 is 36: for example at [9551, 9587]. The maximum prime interval in primes up to 100000 is 72: for example at [31397, 31469]. The maximum prime interval in primes up to 1000000 is 114: for example at [492113, 492227]. The maximum prime interval in primes up to 10000000 is 154: for example at [4652353, 4652507]. The maximum prime interval in primes up to 100000000 is 220: for example at [47326693, 47326913]. The maximum prime interval in primes up to 1000000000 is 282: for example at [436273009, 436273291]. The maximum prime interval in primes up to 10000000000 is 354: for example at [4302407359, 4302407713].
Pascal
Free Pascal
<lang pascal>program primesieve; // sieving small ranges of 65536 only odd numbers //{$O+,R+} {$IFDEF FPC}
{$MODE DELPHI}{$OPTIMIZATION ON,ALL}{$CODEALIGN proc=16}
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;
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 Word;
{$Align 16}
Limit,OffSet,Dekalimit,MaxGap : 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
i := 1;// starts with pr = 3
umf := 1;
repeat
IF preSieve[i] =0 then
Begin
pr := 2*i+1;
j := i;
repeat
preSieve[j] := 1;
inc(j,pr);
until j> High(preSieve);
umf := umf*pr;
end;
inc(i);
until umf>High(preSieve);
preSieveOffset := 0;
end;
procedure CalcSievePrimOfs(lmt:NativeUint); //lmt High(sievePrimes) 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] = 1 do
inc(i);
pr := 2*i+1;
inc(i);
j := ((pr*pr)-1) DIV 2;
if j > High(Sieve) then
BREAK;
repeat
Sieve[j] := 1;
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 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 : pWord; 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,1-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] := 1;
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] := 1;
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,GapPrime : Int64; i: Int32;
begin
T0 := GetTickCount64; Limit := 10*1000*1000;//99999999999;// preSieveInit; sievePrimesInit;
InitSieve;
offset := 0;
Cnt := 1;//==2
LastPrime := 2;
Dekalimit := 10;
MaxGap := 0;
writeln('Limit Gap First prime');
repeat
CopyPreSieveInSieve;sieveOneSieve;CollectPrimes;
inc(Cnt,FoundPrimesCnt);
inc(SieveNum);
//check for max gap
i := 0;
repeat
ActPrime := Offset+FoundPrimes[i];
If ActPrime > Dekalimit then
Begin
writeln(Dekalimit :12,MaxGap:4,GapPrime:12);
DekaLimit := 10 *DekaLimit;
end;
if ActPrime - LastPrime > MaxGap then
begin
MaxGap := ActPrime - LastPrime;
GapPrime:= LastPrime;
end;
LastPrime := ActPrime;
inc(i);
until (i >= FoundPrimesCnt);
inc(offset,2*SieveSize); until SieveNum > (Limit DIV (2*SieveSize));
T1 := GetTickCount64;
OffSet := Uint64(SieveNum-1)*(2*SieveSize);
// correct count to Limit
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. </lang>
- @TIO.RUN:
Limit Gap First prime
10 2 3
100 8 89
1000 20 887
10000 36 9551
100000 72 31397
1000000 114 492113
10000000 154 4652353
100000000 220 47326693
1000000000 282 436273009
10000000000 354 4302407359
455052511 in 10000000000 takes 14319 ms
Perl
<lang perl>use strict; use warnings; use Primesieve qw(generate_primes);
for my $n (2..8) {
my @primes = generate_primes (1, 10**$n);
my($max,$p,$diff) = 0;
map { ($diff = $primes[$_] - $primes[$_-1]) > $max and ($max,$p) = ($diff,$_-1) } 1..$#primes;
printf "Largest prime gap up to %d: %d - between %d and %d.\n", 10**$n, $max, @primes[$p,$p+1];
}</lang>
- Output:
Largest prime gap up to 100: 8 - between 89 and 97. Largest prime gap up to 1000: 20 - between 887 and 907. Largest prime gap up to 10000: 36 - between 9551 and 9587. Largest prime gap up to 100000: 72 - between 31397 and 31469. Largest prime gap up to 1000000: 114 - between 492113 and 492227. Largest prime gap up to 10000000: 154 - between 4652353 and 4652507. Largest prime gap up to 100000000: 220 - between 47326693 and 47326913.
Phix
with javascript_semantics atom t0 = time() constant limit = iff(platform()=JS?1e8:1e9) - 1 sequence primes = get_primes_le(limit) integer maxI = 0, maxDiff = 0 atom nextStop = 10 printf(1,"The largest differences between adjacent primes under the following limits is:\n") for i=2 to length(primes) do integer diff = primes[i] - primes[i-1] if diff>maxDiff then maxDiff = diff maxI = i end if if i==length(primes) or primes[i+1]>nextStop then printf(1,"Under %,d: %,d - %,d = %,d\n", {nextStop, primes[maxI], primes[maxI-1], maxDiff}) nextStop *= 10 end if end for ?elapsed(time()-t0)
- Output:
The largest differences between adjacent primes under the following limits is: Under 10: 5 - 3 = 2 Under 100: 97 - 89 = 8 Under 1,000: 907 - 887 = 20 Under 10,000: 9,587 - 9,551 = 36 Under 100,000: 31,469 - 31,397 = 72 Under 1,000,000: 492,227 - 492,113 = 114 Under 10,000,000: 4,652,507 - 4,652,353 = 154 Under 100,000,000: 47,326,913 - 47,326,693 = 220 Under 1,000,000,000: 436,273,291 - 436,273,009 = 282 "12.7s"
Almost all the time is spent constructing the list of primes. It is about 4 times slower under pwa/p2js so limited to 1e8 when running in a browser to keep the time below 5s instead of over 50s.
Python
<lang python> print("working...") limit = 1000000 res1 = 0 res2 = 0 maxOld = 0 newDiff = 0 oldDiff = 0
def isPrime(n):
for i in range(2,int(n**0.5)+1):
if n%i==0:
return False
return True
for n in range(limit):
newDiff = n - maxOld
if isprime(n):
if (newDiff > oldDiff):
res1 = n
res2 = maxOld
oldDiff = newDiff
maxOld = n
diff = res1 - res2 print(res1) print(res2) print("Largest difference is = ",end="") print(diff) print("done...") </lang>
- Output:
492227 492113 Largest difference is = 114
Raku
Built-ins
<lang perl6>for 2..8 -> $n {
printf "Largest prime gap up to {10 ** $n}: %d - between %d and %d.\n", .[0], |.[1]
given max (^10**$n).grep(&is-prime).rotor(2=>-1).map({.[1]-.[0],$_})
}</lang>
- Output:
Largest prime gap up to 100: 8 - between 89 and 97. Largest prime gap up to 1000: 20 - between 887 and 907. Largest prime gap up to 10000: 36 - between 9551 and 9587. Largest prime gap up to 100000: 72 - between 31397 and 31469. Largest prime gap up to 1000000: 114 - between 492113 and 492227. Largest prime gap up to 10000000: 154 - between 4652353 and 4652507. Largest prime gap up to 100000000: 220 - between 47326693 and 47326913.
Or, significantly faster using a
Module
<lang perl6>use Math::Primesieve; my $sieve = Math::Primesieve.new;
for 2..8 -> $n {
printf "Largest prime gap up to {10 ** $n}: %d - between %d and %d.\n", .[0], |.[1]
given max $sieve.primes(10 ** $n).rotor(2=>-1).map({.[1]-.[0],$_})
}</lang> Same output
Ring
<lang ring> load "stdlib.ring" see "working..." + nl limit = 1000000 Primes = [] maxOld = 0 newDiff = 0 oldDiff = 0
for n = 1 to limit
newDiff = n - maxOld
if isprime(n)
if newDiff > oldDiff
add(Primes,[n,maxOld])
oldDiff = newDiff
ok
maxOld = n
ok
next
len = len(Primes) diff = Primes[len][1] - Primes[len][2] see Primes[len(Primes)] see nl + "Largest difference is = " + diff + nl see "done..." + nl </lang>
- Output:
working... 492227 492113 Largest difference is = 114 done...
Wren
<lang ecmascript>import "./math" for Int import "/fmt" for Fmt
var limit = 1e9 - 1 var primes = Int.primeSieve(limit) var maxI = 0 var maxDiff = 0 var nextStop = 10 System.print("The largest differences between adjacent primes under the following limits is:") for (i in 1...primes.count) {
var diff = primes[i] - primes[i-1]
if (diff > maxDiff) {
maxDiff = diff
maxI = i
}
if (i == primes.count - 1 || primes[i+1] > nextStop) {
Fmt.print("Under $,d: $,d - $,d = $,d", nextStop, primes[maxI], primes[maxI-1], maxDiff)
nextStop = nextStop * 10
}
}</lang>
- Output:
The largest differences between adjacent primes under the following limits is: Under 10: 5 - 3 = 2 Under 100: 97 - 89 = 8 Under 1,000: 907 - 887 = 20 Under 10,000: 9,587 - 9,551 = 36 Under 100,000: 31,469 - 31,397 = 72 Under 1,000,000: 492,227 - 492,113 = 114 Under 10,000,000: 4,652,507 - 4,652,353 = 154 Under 100,000,000: 47,326,913 - 47,326,693 = 220 Under 1,000,000,000: 436,273,291 - 436,273,009 = 282
XPL0
<lang XPL0>def Size = 1_000_000_000/2; \sieve for odd numbers int Prime, I, K; char Flags; int Limit, TenToThe, Max, Gap, I0, Prime0, Prime1; [Flags:= MAlloc(Size+1); \(heap only provides 64 MB) for I:= 0 to Size do \set flags all true
Flags(I):= true;
for I:= 0 to Size do
if Flags(I) then \found a prime
[Prime:= I+I+3;
K:= I+Prime; \first multiple to strike off
while K <= Size do \strike off all multiples
[Flags(K):= false;
K:= K+Prime;
];
];
Text(0, "Largest difference between adjacent primes under: "); Limit:= 10; for TenToThe:= 1 to 9 do
[Max:= 0; I0:= 0;
for I:= 0 to Limit/2-3 do
[if Flags(I) then \odd prime
[Gap:= I - I0;
if Gap > Max then
[Max:= Gap;
Prime0:= I0*2 + 3;
Prime1:= I *2 + 3;
];
I0:= I;
];
];
IntOut(0, Limit); Text(0, " is ");
IntOut(0, Prime1); Text(0, " - ");
IntOut(0, Prime0); Text(0, " = ");
IntOut(0, Max*2); CrLf(0);
Limit:= Limit*10;
];
]</lang>
- Output:
Largest difference between adjacent primes under: 10 is 5 - 3 = 2 100 is 97 - 89 = 8 1000 is 907 - 887 = 20 10000 is 9587 - 9551 = 36 100000 is 31469 - 31397 = 72 1000000 is 492227 - 492113 = 114 10000000 is 4652507 - 4652353 = 154 100000000 is 47326913 - 47326693 = 220 1000000000 is 436273291 - 436273009 = 282