Find adjacent primes which differ by a square integer
- Task
Find adjacent primes under 1,000,000 whose difference (> 36) is a square integer.
11l
F primes_upto(limit)
V is_prime = [0B] * 2 [+] [1B] * (limit - 1)
L(n) 0 .< Int(limit ^ 0.5 + 1.5)
I is_prime[n]
L(i) (n * n .. limit).step(n)
is_prime[i] = 0B
R enumerate(is_prime).filter((i, prime) -> prime).map((i, prime) -> i)
V primes = primes_upto(1'000'000)
F is_square(x)
R Int(sqrt(x)) ^ 2 == x
L(n) 2 .< primes.len
V pr1 = primes[n]
V pr2 = primes[n - 1]
V diff = pr1 - pr2
I (is_square(diff) & diff > 36)
print(pr1‘ ’pr2‘ diff = ’diff)
- Output:
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
ALGOL 68
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
- 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
Arturo
squares: map 7..15 'x -> x*x
primes: select 1..1000000 => prime?
loop.with:'i primes\[0..(size primes)-2] 'p [
next: primes\[i+1]
if contains? squares next-p ->
print [pad to :string next 6 "-" pad to :string p 6 "=" next-p]
]
- 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
AWK
# syntax: GAWK -f FIND_ADJACENTS_PRIMES_WHICH_DIFFERENCE_IS_SQUARE_INTEGER.AWK
# converted from FreeBASIC
BEGIN {
start = i = 3
stop = 999999
while (j <= stop) {
j = next_prime(i)
if (j-i > 36 && is_square(j-i)) {
printf("%9d %9d %9d\n",i,j,j-i)
count++
}
i = j
}
printf("Adjacent primes which difference is square integer (>36) %d-%d: %d\n",start,stop,count)
exit(0)
}
function is_prime(n, d) {
d = 5
if (n < 2) { return(0) }
if (n % 2 == 0) { return(n == 2) }
if (n % 3 == 0) { return(n == 3) }
while (d*d <= n) {
if (n % d == 0) { return(0) }
d += 2
if (n % d == 0) { return(0) }
d += 4
}
return(1)
}
function is_square(n) {
return (int(sqrt(n))^2 == n)
}
function next_prime(n, q) { # finds next prime after n
if (n == 0) { return(2) }
if (n < 3) { return(++n) }
q = n + 2
while (!is_prime(q)) {
q += 2
}
return(q)
}
- 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 Adjacent primes which difference is square integer (>36) 3-999999: 26
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;
}
CLU
% Integer square root
isqrt = proc (s: int) returns (int)
x0: int := s/2
if x0=0 then return(s) end
x1: int := (x0 + s/x0)/2
while x1 < x0 do
x0 := x1
x1 := (x0 + s/x0)/2
end
return(x0)
end isqrt
% See if a number is square
% Note that all squares are 0, 1, 4, or 9 mod 16.
is_square = proc (n: int) returns (bool)
d: int := n//16
if d=0 cor d=1 cor d=4 cor d=9 then
return(n = isqrt(n)**2)
else
return(false)
end
end is_square
% Find all primes up to a given number
sieve = proc (top: int) returns (array[int])
prime: array[bool] := array[bool]$fill(2,top-1,true)
for p: int in int$from_to(2,isqrt(top)) do
if prime[p] then
for c: int in int$from_to_by(p*p,top,p) do
prime[c] := false
end
end
end
list: array[int] := array[int]$predict(1,isqrt(top))
for p: int in int$from_to(2,top) do
if prime[p] then array[int]$addh(list,p) end
end
return(list)
end sieve
start_up = proc ()
MAX = 1000000
DIFF = 36
po: stream := stream$primary_output()
primes: array[int] := sieve(MAX)
for i: int in int$from_to(array[int]$low(primes)+1,
array[int]$high(primes)) do
d: int := primes[i] - primes[i-1]
if d>DIFF cand is_square(d) then
stream$putright(po, int$unparse(primes[i]), 6)
stream$puts(po, " - ")
stream$putright(po, int$unparse(primes[i-1]), 6)
stream$puts(po, " = ")
stream$putright(po, int$unparse(d), 4)
stream$puts(po, " = ")
stream$putright(po, int$unparse(isqrt(d)), 4)
stream$putl(po, "^2")
end
end
end start_up
- Output:
89753 - 89689 = 64 = 8^2 107441 - 107377 = 64 = 8^2 288647 - 288583 = 64 = 8^2 368021 - 367957 = 64 = 8^2 381167 - 381103 = 64 = 8^2 396833 - 396733 = 100 = 10^2 400823 - 400759 = 64 = 8^2 445427 - 445363 = 64 = 8^2 623171 - 623107 = 64 = 8^2 625763 - 625699 = 64 = 8^2 637067 - 637003 = 64 = 8^2 710777 - 710713 = 64 = 8^2 725273 - 725209 = 64 = 8^2 779477 - 779413 = 64 = 8^2 801947 - 801883 = 64 = 8^2 803813 - 803749 = 64 = 8^2 821741 - 821677 = 64 = 8^2 832583 - 832519 = 64 = 8^2 838349 - 838249 = 100 = 10^2 844841 - 844777 = 64 = 8^2 883871 - 883807 = 64 = 8^2 912167 - 912103 = 64 = 8^2 919511 - 919447 = 64 = 8^2 954827 - 954763 = 64 = 8^2 981887 - 981823 = 64 = 8^2 997877 - 997813 = 64 = 8^2
Delphi
function IsPrime(N: integer): boolean;
{Optimised prime test - about 40% faster than the naive approach}
var I,Stop: integer;
begin
if (N = 2) or (N=3) then Result:=true
else if (n <= 1) or ((n mod 2) = 0) or ((n mod 3) = 0) then Result:= false
else
begin
I:=5;
Stop:=Trunc(sqrt(N));
Result:=False;
while I<=Stop do
begin
if ((N mod I) = 0) or ((N mod (i + 2)) = 0) then exit;
Inc(I,6);
end;
Result:=True;
end;
end;
function GetNextPrime(Start: integer): integer;
{Get the next prime number after Start}
begin
repeat Inc(Start)
until IsPrime(Start);
Result:=Start;
end;
procedure ShowPrimeDiffs(Memo: TMemo);
var P1,P2,D: integer;
begin
P1:=GetNextPrime(2);
repeat
begin
P2:=GetNextPrime(P1);
D:=P2 - P1;
if (D>36) and (Frac(sqrt(D))=0) then
begin
Memo.Lines.Add(IntToStr(P2)+' - '+IntToStr(P1)+' = '+IntToStr(D));
end;
P1:=P2;
end
until P2>=1000000;
end;
- 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
EasyLang
fastfunc isprim num .
i = 2
while i <= sqrt num
if num mod i = 0
return 0
.
i += 1
.
return 1
.
prim = 2
proc nextprim . .
repeat
prim += 1
until isprim prim = 1
.
.
func is_square n .
h = floor sqrt n
return if h * h = n
.
while prim < 1000000
prev = prim
nextprim
if prim - prev > 36 and is_square (prim - prev) = 1
print prim & " - " & prev & " = " & prim - prev
.
.
- 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
F#
This task uses Extensible Prime Generator (F#)
// 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")
- 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)
Factor
USING: formatting io kernel lists lists.lazy math math.functions
math.primes.lists sequences ;
: adj-primes ( -- list ) lprimes dup cdr lzip ;
: diff ( pair -- n ) first2 swap - ;
: adj-primes-diff ( -- list )
adj-primes [ dup diff suffix ] lmap-lazy ;
: big-adj-primes-diff ( -- list )
adj-primes-diff [ last 36 > ] lfilter ;
: square? ( n -- ? ) sqrt dup >integer number= ;
: big-sq-adj-primes-diff ( -- list )
big-adj-primes-diff [ last square? ] lfilter ;
"Adjacent primes under a million whose difference is a square > 36:" print nl
"p1 p2 difference" print
"============================" print
big-sq-adj-primes-diff [ second 1,000,000 < ] lwhile
[ "%-6d %-6d %d\n" vprintf ] leach
- Output:
Adjacent primes under a million whose difference is a square > 36: p1 p2 difference ============================ 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
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;
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
- 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
Go
package main
import (
"fmt"
"math"
"rcu"
)
func main() {
limit := 999999
primes := rcu.Primes(limit)
fmt.Println("Adjacent primes under 1,000,000 whose difference is a square > 36:")
for i := 1; i < len(primes); i++ {
diff := primes[i] - primes[i-1]
if diff > 36 {
s := int(math.Sqrt(float64(diff)))
if diff == s*s {
cp1 := rcu.Commatize(primes[i])
cp2 := rcu.Commatize(primes[i-1])
fmt.Printf("%7s - %7s = %3d = %2d x %2d\n", cp1, cp2, diff, s, s)
}
}
}
}
- Output:
Same as Wren example.
GW-BASIC
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
Haskell
import Data.List.Split ( divvy )
isSquare :: Int -> Bool
isSquare n = (snd $ properFraction $ sqrt $ fromIntegral n) == 0.0
isPrime :: Int -> Bool
isPrime n
|n == 2 = True
|n == 1 = False
|otherwise = null $ filter (\i -> mod n i == 0 ) [2 .. root]
where
root :: Int
root = floor $ sqrt $ fromIntegral n
solution :: [[Int]]
solution = filter (\li -> isSquare (last li - head li ) &&
( last li - head li ) > 36 ) $ divvy 2 1 $ filter isPrime [2..1000000]
printResultLine :: [Int] -> String
printResultLine list = show ( last list ) ++ " - " ++ ( show $ head list )
++ " = " ++ ( show ( last list - head list ))
main :: IO ( )
main = do
let resultPairs = solution
mapM_ (\li -> putStrLn $ printResultLine li ) resultPairs
- 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
J
#(,.-~/"1) p:0 1+/~I.(= <.)6.5>.%:2-~/\p:i.p:inv 1e6 NB. count them
26
(,.-~/"1) p:0 1+/~I.(= <.)6.5>.%:2-~/\p:i.p:inv 1e6 NB. show them
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
In other words: enumerate primes less than 1e6, find the pairwise differences, find where the prime pairs where maximum of their square root and 6.5 is an integer, and list those pairs with their differences.
jq
Works with gojq, the Go implementation of jq
See Erdős-primes#jq for a suitable definition of `is_prime` as used here.
Preliminaries
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
# Primes less than . // infinite
def primes:
(. // infinite) as $n
| if $n < 3 then empty
else 2, (range(3; $n) | select(is_prime))
end;
The task
# Input is given to primes/0 - to determine the maximum prime to consider
# Output: stream of [$prime, $nextPrime]
def adjacentPrimesWhichDifferBySquare:
def isSquare: sqrt | . == floor;
foreach primes as $p ( {previous: null};
.emit = null
| if .previous != null
and (($p - .previous) | isSquare)
then .emit = [.previous, $p]
else .
end
| .previous = $p;
select(.emit).emit);
# Input is given to primes/0 to determine the maximum prime to consider.
# Gap must be greater than $gap
def task($gap):
def l: lpad(6);
"Adjacent primes under \(.) whose difference is a square > \($gap):",
(adjacentPrimesWhichDifferBySquare
| (.[1] - .[0]) as $diff
| select($diff > $gap)
| "\(.[1]|l) - \(.[0]|l) = \($diff|lpad(4))" ) ;
1E6 | task(36)
- Output:
As for #ALGOL_68.
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)
- 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).
Mathematica /Wolfram Language
ps = Prime[Range[PrimePi[10^6]]];
ps = Partition[ps, 2, 1];
ps = {#1, #2, #2 - #1} & @@@ ps;
ps //= Select[Extract[{3}]/*GreaterThan[36]];
ps //= Select[Extract[{3}]/*Sqrt/*IntegerQ];
ps // Grid
- 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
PARI/GP
for(i=3,1000000,j=nextprime(i+1);if(isprime(i)&&j-i>36&&issquare(j-i),print(i," ",j," ",j-i)))
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] - $primeref->[$i - 1] = $diff\n";
}
- 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
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
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...")
- 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...
Quackery
eratosthenes
, isprime
, and sqrt
are defined at Sieve of Eratosthenes#Quackery.
1000000 eratosthenes
0 0
1000000 times
[ i^ isprime if
[ nip i^ 2dup swap -
dup 36 > iff
[ dup sqrt dup * = if
[ 2dup swap
2dup - unrot
echo say " - "
echo say " = "
echo cr ] ]
else drop ] ]
2drop
- 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
Raku
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(', ');
}
- 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
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
- 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...
Ruby
require "prime"
Prime.each(1_000_000).each_cons(2) do |a, b|
diff = b - a
next unless diff > 36
isqrt = Integer.sqrt(diff)
puts "#{b} - #{a} = #{diff}" if isqrt*isqrt == diff
end
- 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
Rust
use prime_tools ;
fn is_square_number( num : u32 ) -> bool {
let comp_num : f32 = num as f32 ;
let root = comp_num.sqrt( ) ;
return root == root.floor( ) ;
}
fn main() {
let primes: Vec<u32> = prime_tools::get_primes_less_than_x(1000000_u32) ;
let len = primes.len( ) ;
let mut i : usize = 0 ;
while i < len - 1 {
let diff : u32 = primes[ i + 1 ] - primes[ i ] ;
if diff > 36 && is_square_number( diff ) {
println!("{} - {} = {}" , primes[ i + 1 ] , primes[ i ] , diff) ;
}
i += 1 ;
}
}
- 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
Sidef
var p = 2
var upto = 1e6
each_prime(p.next_prime, upto, {|q|
if (q-p > 36 && is_square(q-p)) {
say "#{'%6s' % q} - #{'%6s' % p} = #{'%2s' % isqrt(q-p)}^2"
}
p = q
})
- Output:
89753 - 89689 = 8^2 107441 - 107377 = 8^2 288647 - 288583 = 8^2 368021 - 367957 = 8^2 381167 - 381103 = 8^2 396833 - 396733 = 10^2 400823 - 400759 = 8^2 445427 - 445363 = 8^2 623171 - 623107 = 8^2 625763 - 625699 = 8^2 637067 - 637003 = 8^2 710777 - 710713 = 8^2 725273 - 725209 = 8^2 779477 - 779413 = 8^2 801947 - 801883 = 8^2 803813 - 803749 = 8^2 821741 - 821677 = 8^2 832583 - 832519 = 8^2 838349 - 838249 = 10^2 844841 - 844777 = 8^2 883871 - 883807 = 8^2 912167 - 912103 = 8^2 919511 - 919447 = 8^2 954827 - 954763 = 8^2 981887 - 981823 = 8^2 997877 - 997813 = 8^2
Wren
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)
}
}
}
- 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
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)
];
]
- 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