Find adjacent primes which differ by a square integer
- Task
Find adjacent primes under 1,000,000 whose difference (> 36) is a square integer.
ALGOL 68
<lang algol68>BEGIN # find a adjacent primes where the primes differ by a square > 36 #
INT min diff = 37; INT max prime = 1 000 000; PR read "primes.incl.a68" PR # form a list of primes to max prime # []INT prime = EXTRACTPRIMESUPTO max prime FROMPRIMESIEVE PRIMESIEVE max prime; # construct a table of squares, we will need at most the square root of max prime # # but in reality much less than that - assume 1000 will be enough # [ 1 : 1000 ]BOOL is square; FOR i TO UPB is square DO is square[ i ] := FALSE OD; FOR i WHILE INT i2 = i * i; i2 <= UPB is square DO is square[ i2 ] := TRUE OD; # find the primes # FOR p TO UPB prime - 1 DO INT q = p + 1; INT diff = prime[ q ] - prime[ p ]; IF diff > min diff AND is square[ diff ] THEN print( ( whole( prime[ q ], -6 ), " - ", whole( prime[ p ], -6 ), " = ", whole( diff, 0 ), newline ) ) FI OD
END</lang>
- Output:
89753 - 89689 = 64 107441 - 107377 = 64 288647 - 288583 = 64 368021 - 367957 = 64 381167 - 381103 = 64 396833 - 396733 = 100 400823 - 400759 = 64 445427 - 445363 = 64 623171 - 623107 = 64 625763 - 625699 = 64 637067 - 637003 = 64 710777 - 710713 = 64 725273 - 725209 = 64 779477 - 779413 = 64 801947 - 801883 = 64 803813 - 803749 = 64 821741 - 821677 = 64 832583 - 832519 = 64 838349 - 838249 = 100 844841 - 844777 = 64 883871 - 883807 = 64 912167 - 912103 = 64 919511 - 919447 = 64 954827 - 954763 = 64 981887 - 981823 = 64 997877 - 997813 = 64
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 issquare( int p ) {
int i; for(i=0;i*i<p;i++); return i*i==p;
}
int main(void) {
int i=3, j=2; for(i=3;j<=1000000;i=j) { j=nextprime(i); if(j-i>36&&issquare(j-i)) printf( "%d %d %d\n", i, j, j-i ); } return 0;
}</lang>
F#
This task uses Extensible Prime Generator (F#) <lang fsharp> // Find adjacents primes which difference is square integer . Nigel Galloway: November 23rd., 2021 primes32()|>Seq.takeWhile((>)1000000)|>Seq.pairwise|>Seq.filter(fun(n,g)->let n=g-n in let g=(float>>sqrt>>int)n in g>6 && n=g*g)|>Seq.iter(printfn "%A") </lang>
- Output:
(89689, 89753) (107377, 107441) (288583, 288647) (367957, 368021) (381103, 381167) (396733, 396833) (400759, 400823) (445363, 445427) (623107, 623171) (625699, 625763) (637003, 637067) (710713, 710777) (725209, 725273) (779413, 779477) (801883, 801947) (803749, 803813) (821677, 821741) (832519, 832583) (838249, 838349) (844777, 844841) (883807, 883871) (912103, 912167) (919447, 919511) (954763, 954827) (981823, 981887) (997813, 997877)
Fermat
<lang fermat>Func Issqr( n ) = if (Sqrt(n))^2=n then 1 else 0 fi.; i:=3; j:=3; while j<1000000 do
j:=i+2; while j < 1000000 do if Isprime(j) then if j-i>36 and Issqr(j-i) then !!(i,j,j-i) fi; i:=j; fi; j:=j+2; od;
od;</lang>
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
function issquare( n as uinteger ) as boolean
if int(sqr(n))^2 = n then return true else return false
end function
dim as uinteger i=3, j=0 while j<1000000
j = nextprime(i) if j-i > 36 and issquare(j-i) then print i, j, j-i i = j
wend</lang>
- Output:
89689 89753 64 107377 107441 64 288583 288647 64 367957 368021 64 381103 381167 64 396733 396833 100 400759 400823 64 445363 445427 64 623107 623171 64 625699 625763 64 637003 637067 64 710713 710777 64 725209 725273 64 779413 779477 64 801883 801947 64 803749 803813 64 821677 821741 64 832519 832583 64 838249 838349 100 844777 844841 64 883807 883871 64 912103 912167 64 919447 919511 64 954763 954827 64 981823 981887 64 997813 997877 64
GW-BASIC
<lang gwbasic>10 P=3 : P2=0 20 GOSUB 180 30 IF P2>1000000! THEN END 40 R = P2-P 50 IF R > 36 AND INT(SQR(R))^2=R THEN PRINT P,P2,R 60 P=P2 70 GOTO 20 80 REM tests if a number is prime 90 Q=0 100 IF P = 2 THEN Q = 1:RETURN 110 IF P=3 THEN Q=1:RETURN 120 I=1 130 I=I+1 140 IF INT(P/I)*I = P THEN RETURN 150 IF I*I<=P THEN GOTO 130 160 Q = 1 170 RETURN 180 REM finds the next prime after P, result in P2 190 IF P = 0 THEN P2 = 2: RETURN 200 IF P<3 THEN P2 = P + 1: RETURN 210 T = P 220 P = P + 1 230 GOSUB 80 240 IF Q = 1 THEN P2 = P: P = T: RETURN 250 GOTO 220</lang>
Julia
<lang julia>using Primes
function squareprimegaps(limit)
pri = primes(limit) squares = Set([1; [x * x for x in 2:2:100]]) diffs = [pri[i] - pri[i - 1] for i in 2:length(pri)] squarediffs = sort(unique(filter(n -> n in squares, diffs))) println("\n\nSquare prime gaps to $limit:") for sq in squarediffs i = findfirst(x -> x == sq, diffs) n = count(x -> x == sq, diffs) if limit == 1000000 && sq > 36 println("Showing all $n with square difference $sq:") pairs = [(pri[i], pri[i + 1]) for i in findall(x -> x == sq, diffs)] foreach(p -> print(last(p), first(p) % 4 == 0 ? "\n" : " "), enumerate(pairs)) else println("Square difference $sq: $n found. Example: ($(pri[i]), $(pri[i + 1])).") end end
end
squareprimegaps(1_000_000) squareprimegaps(10_000_000_000)
</lang>
- Output:
Square prime gaps to 1000000: Square difference 1: 1 found. Example: (2, 3). Square difference 4: 8143 found. Example: (7, 11). Square difference 16: 2881 found. Example: (1831, 1847). Square difference 36: 767 found. Example: (9551, 9587). Showing all 24 with square difference 64: (89689, 89753) (107377, 107441) (288583, 288647) (367957, 368021) (381103, 381167) (400759, 400823) (445363, 445427) (623107, 623171) (625699, 625763) (637003, 637067) (710713, 710777) (725209, 725273) (779413, 779477) (801883, 801947) (803749, 803813) (821677, 821741) (832519, 832583) (844777, 844841) (883807, 883871) (912103, 912167) (919447, 919511) (954763, 954827) (981823, 981887) (997813, 997877) Showing all 2 with square difference 100: (396733, 396833) (838249, 838349) Square prime gaps to 10000000000: Square difference 1: 1 found. Example: (2, 3). Square difference 4: 27409998 found. Example: (7, 11). Square difference 16: 15888305 found. Example: (1831, 1847). Square difference 36: 11593345 found. Example: (9551, 9587). Square difference 64: 1434957 found. Example: (89689, 89753). Square difference 100: 268933 found. Example: (396733, 396833). Square difference 144: 35563 found. Example: (11981443, 11981587). Square difference 196: 1254 found. Example: (70396393, 70396589). Square difference 256: 41 found. Example: (1872851947, 1872852203).
PARI/GP
<lang parigp> for(i=3,1000000,j=nextprime(i+1);if(isprime(i)&&j-i>36&&issquare(j-i),print(i," ",j," ",j-i))) </lang>
Perl
<lang perl>#!/usr/bin/perl
use strict; # https://rosettacode.org/wiki/Find_adjacents_primes_which_difference_is_square_integer use warnings; use ntheory qw( primes is_square );
my $primeref = primes( 1e6 ); for my $i (1 .. $#$primeref )
{ (my $diff = $primeref->[$i] - $primeref->[$i - 1]) > 36 or next; is_square($diff) and print "$primeref->[$i + 1] - $primeref->[$i - 1] = $diff\n"; }</lang>
- Output:
89759 - 89689 = 64 107449 - 107377 = 64 288649 - 288583 = 64 368029 - 367957 = 64 381169 - 381103 = 64 396871 - 396733 = 100 400837 - 400759 = 64 445433 - 445363 = 64 623209 - 623107 = 64 625777 - 625699 = 64 637073 - 637003 = 64 710779 - 710713 = 64 725293 - 725209 = 64 779489 - 779413 = 64 801949 - 801883 = 64 803819 - 803749 = 64 821747 - 821677 = 64 832591 - 832519 = 64 838351 - 838249 = 100 844847 - 844777 = 64 883877 - 883807 = 64 912173 - 912103 = 64 919519 - 919447 = 64 954829 - 954763 = 64 981889 - 981823 = 64 997879 - 997813 = 64
Phix
with javascript_semantics constant limit = 1_000_000 sequence primes = get_primes_le(limit), square = repeat(false,floor(sqrt(limit))) integer sq = 7 while sq*sq<=length(square) do square[sq*sq] = true sq += 1 end while for i=2 to length(primes) do integer p = primes[i], q = primes[i-1], d = p-q if square[d] then printf(1,"%6d - %6d = %d\n",{p,q,d}) end if end for
- Output:
89753 - 89689 = 64 107441 - 107377 = 64 288647 - 288583 = 64 368021 - 367957 = 64 381167 - 381103 = 64 396833 - 396733 = 100 400823 - 400759 = 64 445427 - 445363 = 64 623171 - 623107 = 64 625763 - 625699 = 64 637067 - 637003 = 64 710777 - 710713 = 64 725273 - 725209 = 64 779477 - 779413 = 64 801947 - 801883 = 64 803813 - 803749 = 64 821741 - 821677 = 64 832583 - 832519 = 64 838349 - 838249 = 100 844841 - 844777 = 64 883871 - 883807 = 64 912167 - 912103 = 64 919511 - 919447 = 64 954827 - 954763 = 64 981887 - 981823 = 64 997877 - 997813 = 64
Python
<lang python> import math print("working...") limit = 1000000 Primes = [] oldPrime = 0 newPrime = 0 x = 0
def isPrime(n):
for i in range(2,int(n**0.5)+1): if n%i==0: return False return True
def issquare(x): for n in range(x): if (x == n*n): return 1 return 0
for n in range(limit):
if isPrime(n): Primes.append(n)
for n in range(2,len(Primes)):
pr1 = Primes[n] pr2 = Primes[n-1] diff = pr1 - pr2 flag = issquare(diff) if (flag == 1 and diff > 36): print(str(pr1) + " " + str(pr2) + " diff = " + str(diff))
print("done...") </lang>
- Output:
working... 89753 89689 diff = 64 107441 107377 diff = 64 288647 288583 diff = 64 368021 367957 diff = 64 381167 381103 diff = 64 396833 396733 diff = 100 400823 400759 diff = 64 445427 445363 diff = 64 623171 623107 diff = 64 625763 625699 diff = 64 637067 637003 diff = 64 710777 710713 diff = 64 725273 725209 diff = 64 779477 779413 diff = 64 801947 801883 diff = 64 803813 803749 diff = 64 821741 821677 diff = 64 832583 832519 diff = 64 838349 838249 diff = 100 844841 844777 diff = 64 883871 883807 diff = 64 912167 912103 diff = 64 919511 919447 diff = 64 954827 954763 diff = 64 981887 981823 diff = 64 997877 997813 diff = 64 done...
Raku
<lang perl6>use Lingua::EN::Numbers; use Math::Primesieve;
my $iterator = Math::Primesieve::iterator.new; my $limit = 1e10; my @squares = (1..30).map: *²; my $last = 2; my @gaps; my @counts;
loop {
my $this = (my $p = $iterator.next) - $last; quietly @gaps[$this].push($last) if +@gaps[$this] < 10; @counts[$this]++; last if $p > $limit; $last = $p;
}
print "Adjacent primes up to {comma $limit.Int} with a gap value that is a perfect square:"; for @gaps.pairs.grep: { (.key ∈ @squares) && .value.defined} -> $p {
my $ten = (@counts[$p.key] > 10) ?? ', (first ten)' !! ; say "\nGap {$p.key}: {comma @counts[$p.key]} found$ten:"; put join "\n", $p.value.batch(5)».map({"($_, {$_+ $p.key})"})».join(', ');
}</lang>
- Output:
Adjacent primes up to 10,000,000,000 with a gap value that is a perfect square: Gap 1: 1 found: (2, 3) Gap 4: 27,409,998 found, (first ten): (7, 11), (13, 17), (19, 23), (37, 41), (43, 47) (67, 71), (79, 83), (97, 101), (103, 107), (109, 113) Gap 16: 15,888,305 found, (first ten): (1831, 1847), (1933, 1949), (2113, 2129), (2221, 2237), (2251, 2267) (2593, 2609), (2803, 2819), (3121, 3137), (3373, 3389), (3391, 3407) Gap 36: 11,593,345 found, (first ten): (9551, 9587), (12853, 12889), (14107, 14143), (15823, 15859), (18803, 18839) (22193, 22229), (22307, 22343), (22817, 22853), (24281, 24317), (27143, 27179) Gap 64: 1,434,957 found, (first ten): (89689, 89753), (107377, 107441), (288583, 288647), (367957, 368021), (381103, 381167) (400759, 400823), (445363, 445427), (623107, 623171), (625699, 625763), (637003, 637067) Gap 100: 268,933 found, (first ten): (396733, 396833), (838249, 838349), (1313467, 1313567), (1648081, 1648181), (1655707, 1655807) (2345989, 2346089), (2784373, 2784473), (3254959, 3255059), (3595489, 3595589), (4047157, 4047257) Gap 144: 35,563 found, (first ten): (11981443, 11981587), (18687587, 18687731), (20024339, 20024483), (20388583, 20388727), (21782503, 21782647) (25507423, 25507567), (27010003, 27010147), (28716287, 28716431), (31515413, 31515557), (32817493, 32817637) Gap 196: 1,254 found, (first ten): (70396393, 70396589), (191186251, 191186447), (208744777, 208744973), (233987851, 233988047), (288568771, 288568967) (319183093, 319183289), (336075937, 336076133), (339408151, 339408347), (345247753, 345247949), (362956201, 362956397) Gap 256: 41 found, (first ten): (1872851947, 1872852203), (2362150363, 2362150619), (2394261637, 2394261893), (2880755131, 2880755387), (2891509333, 2891509589) (3353981623, 3353981879), (3512569873, 3512570129), (3727051753, 3727052009), (3847458487, 3847458743), (4008610423, 4008610679)
Ring
<lang ring> load "stdlib.ring" see "working..." + nl limit = 1000000 Primes = [] oldPrime = 0 newPrime = 0 x = 0
for n = 1 to limit
if isprime(n) add(Primes,n) ok
next
for n = 2 to len(Primes)
pr1 = Primes[n] pr2 = Primes[n-1] diff = pr1 - pr2 flag = issquare(diff) if flag = 1 and diff > 36 see "" + pr1 + " " + pr2 + " diff = " + diff + nl ok
next
see "done..." + nl
func issquare(x)
for n = 1 to sqrt(x) if x = pow(n,2) return 1 ok next return 0
</lang>
- Output:
working... 89753 89689 diff = 64 107441 107377 diff = 64 288647 288583 diff = 64 368021 367957 diff = 64 381167 381103 diff = 64 396833 396733 diff = 100 400823 400759 diff = 64 445427 445363 diff = 64 623171 623107 diff = 64 625763 625699 diff = 64 637067 637003 diff = 64 710777 710713 diff = 64 725273 725209 diff = 64 779477 779413 diff = 64 801947 801883 diff = 64 803813 803749 diff = 64 821741 821677 diff = 64 832583 832519 diff = 64 838349 838249 diff = 100 844841 844777 diff = 64 883871 883807 diff = 64 912167 912103 diff = 64 919511 919447 diff = 64 954827 954763 diff = 64 981887 981823 diff = 64 997877 997813 diff = 64 done...
Wren
<lang ecmascript>import "./math" for Int import "./fmt" for Fmt
var limit = 1e6 - 1 var primes = Int.primeSieve(limit) System.print("Adjacent primes under 1,000,000 whose difference is a square > 36:") for (i in 1...primes.count) {
var diff = primes[i] - primes[i-1] if (diff > 36) { var s = diff.sqrt.floor if (diff == s * s) { Fmt.print ("$,7d - $,7d = $3d = $2d x $2d", primes[i], primes[i-1], diff, s, s) } }
}</lang>
- Output:
Adjacent primes under 1,000,000 whose difference is a square > 36: 89,753 - 89,689 = 64 = 8 x 8 107,441 - 107,377 = 64 = 8 x 8 288,647 - 288,583 = 64 = 8 x 8 368,021 - 367,957 = 64 = 8 x 8 381,167 - 381,103 = 64 = 8 x 8 396,833 - 396,733 = 100 = 10 x 10 400,823 - 400,759 = 64 = 8 x 8 445,427 - 445,363 = 64 = 8 x 8 623,171 - 623,107 = 64 = 8 x 8 625,763 - 625,699 = 64 = 8 x 8 637,067 - 637,003 = 64 = 8 x 8 710,777 - 710,713 = 64 = 8 x 8 725,273 - 725,209 = 64 = 8 x 8 779,477 - 779,413 = 64 = 8 x 8 801,947 - 801,883 = 64 = 8 x 8 803,813 - 803,749 = 64 = 8 x 8 821,741 - 821,677 = 64 = 8 x 8 832,583 - 832,519 = 64 = 8 x 8 838,349 - 838,249 = 100 = 10 x 10 844,841 - 844,777 = 64 = 8 x 8 883,871 - 883,807 = 64 = 8 x 8 912,167 - 912,103 = 64 = 8 x 8 919,511 - 919,447 = 64 = 8 x 8 954,827 - 954,763 = 64 = 8 x 8 981,887 - 981,823 = 64 = 8 x 8 997,877 - 997,813 = 64 = 8 x 8
XPL0
<lang XPL0>func IsPrime(N); \Return 'true' if odd N > 2 is prime int N, I; [for I:= 3 to sqrt(N) do
[if rem(N/I) = 0 then return false; I:= I+1; ];
return true; ];
int N, P0, P1, D, RD; [P0:= 2; for N:= 3 to 1_000_000-1 do
[if IsPrime(N) then [P1:= N; D:= P1 - P0; \D is even because odd - odd = even if D >= 64 then \the next even square > 36 is 64 [RD:= sqrt(D); if RD*RD = D then [IntOut(0, P1); Text(0, " - "); IntOut(0, P0); Text(0, " = "); IntOut(0, D); CrLf(0); ]; ]; P0:= P1; ]; N:= N+1; \step by 1+1 = 2 (for odd numbers) ];
]</lang>
- Output:
89753 - 89689 = 64 107441 - 107377 = 64 288647 - 288583 = 64 368021 - 367957 = 64 381167 - 381103 = 64 396833 - 396733 = 100 400823 - 400759 = 64 445427 - 445363 = 64 623171 - 623107 = 64 625763 - 625699 = 64 637067 - 637003 = 64 710777 - 710713 = 64 725273 - 725209 = 64 779477 - 779413 = 64 801947 - 801883 = 64 803813 - 803749 = 64 821741 - 821677 = 64 832583 - 832519 = 64 838349 - 838249 = 100 844841 - 844777 = 64 883871 - 883807 = 64 912167 - 912103 = 64 919511 - 919447 = 64 954827 - 954763 = 64 981887 - 981823 = 64 997877 - 997813 = 64