# Factorial primes

A factorial prime is a prime number that is one less or one more than a factorial.

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

In other words a non-negative integer n corresponds to a factorial prime if either n! - 1 or n! + 1 is prime.

Examples

4 corresponds to the factorial prime 4! - 1 = 23.

5 doesn't correspond to a factorial prime because neither 5! - 1 = 119 (7 x 17) nor 5! + 1 = 121 (11 x 11) are prime.

Find and show here the first 10 factorial primes. As well as the prime itself show the factorial number n to which it corresponds and whether 1 is to be added or subtracted.

As 0! (by convention) and 1! are both 1, ignore the former and start counting from 1!.

Stretch

If your language supports arbitrary sized integers, do the same for at least the next 19 factorial primes.

As it can take a long time to demonstrate that a large number (above say 2^64) is definitely prime, you may instead use a function which shows that a number is probably prime to a reasonable degree of certainty. Most 'big integer' libraries have such a function.

If a number has more than 40 digits, do not show the full number. Show instead the first 20 and the last 20 digits and how many digits in total the number has.

Reference

## ALGOL 68

Basic task. Assumes LONG INT is at least 64 bits.

```BEGIN # find some factorial primes - primes that are f - 1 or f + 1      #
#      for some factorial f                                        #

# is prime PROC based on the one in the primality by trial division task #
PROC is prime = ( LONG INT p )BOOL:
IF p <= 1 OR NOT ODD p THEN
p = 2
ELSE
BOOL prime := TRUE;
FOR i FROM 3 BY 2 TO SHORTEN ENTIER long sqrt(p) WHILE prime := p MOD i /= 0 DO SKIP OD;
prime
FI;
# end of code based on the primality by trial division task              #

LONG INT f        := 1;
INT      fp count := 0;
FOR n WHILE fp count < 10 DO
f *:= n;
CHAR fp op := "-";
FOR offset FROM -1 BY 2 TO 1 DO
IF  LONG INT fp = f + offset;
is prime( fp )
THEN
print( ( whole( fp count +:= 1, -2 ), ":", whole( n, -4 )
, "! ", fp op, " 1 = ", whole( fp, 0 )
, newline
)
)
FI;
fp op := "+"
OD
OD
END```
Output:
``` 1:   1! + 1 = 2
2:   2! + 1 = 3
3:   3! - 1 = 5
4:   3! + 1 = 7
5:   4! - 1 = 23
6:   6! - 1 = 719
7:   7! - 1 = 5039
8:  11! + 1 = 39916801
9:  12! - 1 = 479001599
10:  14! - 1 = 87178291199
```

## Arturo

```found: 0
i: 1

while [found < 10][
fct: factorial i

if prime? dec fct [
found: found + 1
print [pad (to :string found) ++ ":" 4 (to :string i)++"! - 1 = " dec fct]
]
if prime? inc fct [
found: found + 1
print [pad (to :string found) ++ ":" 4 (to :string i)++"! + 1 = " inc fct]
]
i: i + 1
]
```
Output:
```  1: 1! + 1 =  2
2: 2! + 1 =  3
3: 3! - 1 =  5
4: 3! + 1 =  7
5: 4! - 1 =  23
6: 6! - 1 =  719
7: 7! - 1 =  5039
8: 11! + 1 =  39916801
9: 12! - 1 =  479001599
10: 14! - 1 =  87178291199```

## BASIC

### BASIC256

Translation of: FreeBASIC
```include "isprime.kbs"
include "factorial.kbs"

print "First 10 factorial primes:"
found = 0
i = 1
while found < 9
fct = factorial (i)

if isprime(fct-1) then
found += 1
print rjust(string(found),2); ": "; rjust(string(i),2); "! - 1 = "; fct-1
end if
if isprime(fct+1) then
found += 1
print rjust(string(found),2); ": "; rjust(string(i),2); "! + 1 = "; fct+1
end if
i += 1
end while
end
```

### Craft Basic

```define found = 0, fct = 0, i = 1

do

if found < 10 then

let fct = factorial(i)

if prime(fct - 1) then

let found = found + 1
print found, ": ", i, "! - 1 = ", fct - 1

endif

if prime(fct + 1) then

let found = found + 1
print found, ": ", i, "! + 1 = ", fct + 1

endif

let i = i + 1

endif

wait

loop found < 10
```

### FreeBASIC

```#include "isprime.bas"
#include "factorial.bas"

Print "First 10 factorial primes:"
Dim As Integer found = 0, i = 1
While found < 10
Dim As Integer fct = factorial (i)

If isprime(fct-1) Then
found += 1
Print Using "##: ##_! - 1 = &"; found; i; fct-1
End If
If isprime(fct+1) Then
found += 1
Print Using "##: ##_! + 1 = &"; found; i; fct+1
End If
i += 1
Wend
Sleep```
Output:
```First 10 factorial primes;
1:  1! + 1 = 2
2:  2! + 1 = 3
3:  3! - 1 = 5
4:  3! + 1 = 7
5:  4! - 1 = 23
6:  6! - 1 = 719
7:  7! - 1 = 5039
8: 11! + 1 = 39916801
9: 12! - 1 = 479001599
10: 14! - 1 = 87178291199```

### Gambas

```Public Sub Main()

Print "First 10 factorial primes:"
Dim found As Long = 0, i As Integer = 1
While found < 10    'más de 10 es ya BigInt
Dim fct As Long = factorial(i)

If isPrime(fct - 1) Then
found += 1
Print Format\$(found, "##"); ": "; Format\$(i, "##"); "! - 1 = "; fct - 1
End If
If isPrime(fct + 1) Then
found += 1
Print Format\$(found, "##"); ": "; Format\$(i, "##"); "! + 1 = "; fct + 1
End If
i += 1
Wend

End

Public Sub isPrime(ValorEval As Long) As Boolean

If ValorEval < 2 Then Return False
If ValorEval Mod 2 = 0 Then Return ValorEval = 2
If ValorEval Mod 3 = 0 Then Return ValorEval = 3
Dim d As Long = 5
While d * d <= ValorEval
If ValorEval Mod d = 0 Then Return False Else d += 2
Wend
Return True

End Function

Public Function factorial(num As Integer) As Long

Dim result As Long = 1
For i As Integer = 2 To num
result *= i
Next
Return result

End
```
Output:
`Same as FreeBASIC entry.`

### PureBasic

```;XIncludeFile "isprime.pb"
;XIncludeFile "factorial.pb"

If OpenConsole()
PrintN("First 10 factorial primes:")
Define found.i = 0, i,i = 1, fct.i

While found < 10
fct = factorial (i)

If isprime(fct-1)
found + 1
PrintN(RSet(Str(found),2) + ": " + RSet(Str(i),2) + "! - 1 = " + Str(fct-1))
EndIf
If isprime(fct+1)
found + 1
PrintN(RSet(Str(found),2) + ": " + RSet(Str(i),2) + "! + 1 = " + Str(fct+1))
EndIf
i + 1
Wend

PrintN(#CRLF\$ + "--- terminado, pulsa RETURN---"): Input()
CloseConsole()
EndIf```
Output:
`Same as FreeBASIC entry.`

### Run BASIC

```function isPrime(n)
if n < 2       then isPrime = 0 : goto [exit]
if n = 2       then isPrime = 1 : goto [exit]
if n mod 2 = 0 then isPrime = 0 : goto [exit]
isPrime = 1
for i = 3 to int(n^.5) step 2
if n mod i = 0 then isPrime = 0 : goto [exit]
next i
[exit]
end function

function factorial(n)
factorial = 1
if n > 1 then factorial = n * factorial(n -1)
end function

print "First 10 factorial primes:"
found = 0
i = 1
while found < 10
fct = factorial(i)

if isPrime(fct-1) then
found = found + 1
print using("##", found); ": "; using("##", i); "! - 1 = "; fct-1
end if
if isPrime(fct+1) then
found = found + 1
print using("##", found); ": "; using("##", i); "! + 1 = "; fct+1
end if
i = i + 1
wend```

### Yabasic

Translation of: FreeBASIC
```import isprime
import factorial

print "First 10 factorial primes:"
found = 0
i = 1
while found < 10
fct = factorial (i)

if isPrime(fct-1) then
found = found + 1
print found using("##"), ": ", i using("##"), "! - 1 = ", fct-1
fi
if isPrime(fct+1) then
found = found + 1
print found using("##"), ": ", i using("##"), "! + 1 = ", fct+1
fi
i = i + 1
end while
```

## C++

```#include <iomanip>
#include <iostream>

#include <gmpxx.h>

using big_int = mpz_class;

std::string to_string(const big_int& num, size_t max_digits) {
std::string str = num.get_str();
size_t len = str.size();
if (len > max_digits) {
str.replace(max_digits / 2, len - max_digits, "...");
str += " (";
str += std::to_string(len);
str += " digits)";
}
return str;
}

bool is_probably_prime(const big_int& n) {
return mpz_probab_prime_p(n.get_mpz_t(), 25) != 0;
}

int main() {
big_int f = 1;
for (int i = 0, n = 1; i < 31; ++n) {
f *= n;
if (is_probably_prime(f - 1)) {
++i;
std::cout << std::setw(2) << i << ": " << std::setw(3) << n
<< "! - 1 = " << to_string(f - 1, 40) << '\n';
}
if (is_probably_prime(f + 1)) {
++i;
std::cout << std::setw(2) << i << ": " << std::setw(3) << n
<< "! + 1 = " << to_string(f + 1, 40) << '\n';
}
}
}
```
Output:
``` 1:   1! + 1 = 2
2:   2! + 1 = 3
3:   3! - 1 = 5
4:   3! + 1 = 7
5:   4! - 1 = 23
6:   6! - 1 = 719
7:   7! - 1 = 5039
8:  11! + 1 = 39916801
9:  12! - 1 = 479001599
10:  14! - 1 = 87178291199
11:  27! + 1 = 10888869450418352160768000001
12:  30! - 1 = 265252859812191058636308479999999
13:  32! - 1 = 263130836933693530167218012159999999
14:  33! - 1 = 8683317618811886495518194401279999999
15:  37! + 1 = 13763753091226345046...79581580902400000001 (44 digits)
16:  38! - 1 = 52302261746660111176...24100074291199999999 (45 digits)
17:  41! + 1 = 33452526613163807108...40751665152000000001 (50 digits)
18:  73! + 1 = 44701154615126843408...03680000000000000001 (106 digits)
19:  77! + 1 = 14518309202828586963...48000000000000000001 (114 digits)
20:  94! - 1 = 10873661566567430802...99999999999999999999 (147 digits)
21: 116! + 1 = 33931086844518982011...00000000000000000001 (191 digits)
22: 154! + 1 = 30897696138473508879...00000000000000000001 (272 digits)
23: 166! - 1 = 90036917057784373664...99999999999999999999 (298 digits)
24: 320! + 1 = 21161033472192524829...00000000000000000001 (665 digits)
25: 324! - 1 = 22889974601791023211...99999999999999999999 (675 digits)
26: 340! + 1 = 51008644721037110809...00000000000000000001 (715 digits)
27: 379! - 1 = 24840307460964707050...99999999999999999999 (815 digits)
28: 399! + 1 = 16008630711655973815...00000000000000000001 (867 digits)
29: 427! + 1 = 29063471769607348411...00000000000000000001 (940 digits)
30: 469! - 1 = 67718096668149510900...99999999999999999999 (1051 digits)
31: 546! - 1 = 14130200926141832545...99999999999999999999 (1260 digits)
```

## Delphi

Works with: Delphi version 6.0

```function IsPrime(N: int64): 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*1.0));
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 Factorial(N: Word): int64;
var  I: integer;
begin
Result:= 1;
for I := 2 to N do Result:=Result * I;
end;

procedure ShowFactorialPrimes(Memo: TMemo);
{Show factorials where F+1 or F-1 are prime}
var I,Cnt: integer;
var F: int64;

procedure DisplayItem(Minus: boolean);
var S: string;
var Sign: char;
var F1: int64;
begin
Inc(Cnt);
if Minus then F1:=F-1 else F1:=F+1;
if Minus then Sign:='-' else Sign:='+';
S:=Format('%2d:   %3d! %s 1 = %d',[Cnt,I,Sign,F1]);
end;

begin
Cnt:=0;
for I:=1 to High(Integer) do
begin
F:=Factorial(I);
if IsPrime(F+1) then DisplayItem(False);
if IsPrime(F-1) then DisplayItem(True);
if Cnt>=10 then break;
end;
end;
```
Output:
``` 1:     1! + 1 = 2
2:     2! + 1 = 3
3:     3! + 1 = 7
4:     3! - 1 = 5
5:     4! - 1 = 23
6:     6! - 1 = 719
7:     7! - 1 = 5039
8:    11! + 1 = 39916801
9:    12! - 1 = 479001599
10:    14! - 1 = 87178291199
```

## EasyLang

Translation of: Lua
```func isprim num .
if num < 2
return 0
.
i = 2
while i <= sqrt num
if num mod i = 0
return 0
.
i += 1
.
return 1
.
f = 1
while count < 10
n += 1
f *= n
op\$ = "-"
for fp in [ f - 1 f + 1 ]
if isprim fp = 1
count += 1
print n & "! " & op\$ & " 1 = " & fp
.
op\$ = "+"
.
.```

## F#

```// Factorial primes. Nigel Galloway: August 15th., 2022
let fN g=if Open.Numeric.Primes.MillerRabin.IsProbablePrime &g then Some(g) else None
let fp()=let rec fG n g=seq{let n=n*g in yield (fN(n-1I),-1,g); yield (fN(n+1I),1,g); yield! fG n (g+1I)} in fG 1I 1I|>Seq.filter(fun(n,_,_)->Option.isSome n)
fp()|>Seq.iteri(fun i (n,g,l)->printfn \$"""%2d{i+1}: %3d{int l}!%+d{g} -> %s{let n=string(Option.get n) in if n.Length<41 then n else n[0..19]+".."+n[n.Length-20..]+" ["+string(n.Length)+" digits]"}""")
```
Output:
``` 1:   1!+1 -> 2
2:   2!+1 -> 3
3:   3!-1 -> 5
4:   3!+1 -> 7
5:   4!-1 -> 23
6:   6!-1 -> 719
7:   7!-1 -> 5039
8:  11!+1 -> 39916801
9:  12!-1 -> 479001599
10:  14!-1 -> 87178291199
11:  27!+1 -> 10888869450418352160768000001
12:  30!-1 -> 265252859812191058636308479999999
13:  32!-1 -> 263130836933693530167218012159999999
14:  33!-1 -> 8683317618811886495518194401279999999
15:  37!+1 -> 13763753091226345046..79581580902400000001 [44 digits]
16:  38!-1 -> 52302261746660111176..24100074291199999999 [45 digits]
17:  41!+1 -> 33452526613163807108..40751665152000000001 [50 digits]
18:  73!+1 -> 44701154615126843408..03680000000000000001 [106 digits]
19:  77!+1 -> 14518309202828586963..48000000000000000001 [114 digits]
20:  94!-1 -> 10873661566567430802..99999999999999999999 [147 digits]
21: 116!+1 -> 33931086844518982011..00000000000000000001 [191 digits]
22: 154!+1 -> 30897696138473508879..00000000000000000001 [272 digits]
23: 166!-1 -> 90036917057784373664..99999999999999999999 [298 digits]
24: 320!+1 -> 21161033472192524829..00000000000000000001 [665 digits]
25: 324!-1 -> 22889974601791023211..99999999999999999999 [675 digits]
26: 340!+1 -> 51008644721037110809..00000000000000000001 [715 digits]
27: 379!-1 -> 24840307460964707050..99999999999999999999 [815 digits]
28: 399!+1 -> 16008630711655973815..00000000000000000001 [867 digits]
29: 427!+1 -> 29063471769607348411..00000000000000000001 [940 digits]
30: 469!-1 -> 67718096668149510900..99999999999999999999 [1051 digits]
31: 546!-1 -> 14130200926141832545..99999999999999999999 [1260 digits]
```

## FutureBasic

```
include "NSLog.incl"

local fn Factorial( n as NSUInteger ) as NSUInteger
NSUInteger factorial = 1

if n > 1 then factorial = n * fn Factorial( n -1 )
end fn = factorial

local fn IsPrime( n as NSUInteger ) as BOOL
BOOL       isPrime = YES
NSUInteger i

if n < 2        then exit fn = NO
if n = 2        then exit fn = YES
if n mod 2 == 0 then exit fn = NO
for i = 3 to int(n^.5) step 2
if n mod i == 0 then exit fn = NO
next
end fn = isPrime

void local fn FactorialPrimes( n as long )
NSUInteger found = 0, i = 1

NSLog( @"First %lu factorial primes:", n )
while ( found < n )
NSUInteger fct = fn Factorial( i )
if ( fn IsPrime( fct - 1 ) )
found++
NSLog( @"%2lu: %3lu! - 1 = %-lu", found, i, fct - 1 )
end if
if ( fn IsPrime( fct + 1 ) )
found++
NSLog( @"%2lu: %3lu! + 1 = %-lu", found, i, fct + 1 )
end if
i++
wend
end fn

fn FactorialPrimes( 10 )

HandleEvents```
Output:
```First 10 factorial primes:
1:   1! + 1 = 2
2:   2! + 1 = 3
3:   3! - 1 = 5
4:   3! + 1 = 7
5:   4! - 1 = 23
6:   6! - 1 = 719
7:   7! - 1 = 5039
8:  11! + 1 = 39916801
9:  12! - 1 = 479001599
10:  14! - 1 = 87178291199
```

## J

```    (,. (-!)/"1)1>.(,. >.@(!inv)@<:) (#~ 1 p: ]) ~.,(!i.27x)+/1 _1
2  1  1
3  2  1
7  3  1
5  3 _1
23  4 _1
719  6 _1
5039  7 _1
39916801 11  1
479001599 12 _1
87178291199 14 _1
```

(i.28x here would have given us an eleventh prime, but the task asked for the first 10, and the stretch goal requires considerable patience.)

## Java

```public class MainApp {
public static void main(String[] args) {
int countOfPrimes = 0;
final int targetCountOfPrimes = 10;
long f = 1;
while (countOfPrimes < targetCountOfPrimes) {
long factorialNum = getFactorial(f);
boolean primePlus = isPrime(factorialNum + 1);
boolean primeMinus = isPrime(factorialNum - 1);
if (primeMinus) {
countOfPrimes++;
System.out.println(countOfPrimes + ": " + factorialNum + "! - 1 = " + (factorialNum - 1));

}
if (primePlus  && f > 1) {
countOfPrimes++;
System.out.println(countOfPrimes + ": " + factorialNum + "! + 1 = " + (factorialNum + 1));
}
f++;
}
}

private static long getFactorial(long f) {
long factorial = 1;
for (long i = 1; i < f; i++) {
factorial *= i;
}
return factorial;
}

private static boolean isPrime(long num) {
if (num < 2) {return false;}
for (long i = 2; i < num; i++) {
if (num % i == 0) {return false;}
}
return true;
}
}
```

Results

```1: 1! + 1 = 2
2: 2! + 1 = 3
3: 6! - 1 = 5
4: 6! + 1 = 7
5: 24! - 1 = 23
6: 720! - 1 = 719
7: 5040! - 1 = 5039
8: 39916800! + 1 = 39916801
9: 479001600! - 1 = 479001599
10: 87178291200! - 1 = 87178291199
```

### Another Way with BigIntegers

```import java.math.BigInteger;

public class MainApp {
public static void main(String[] args) {
//Used to measure total runtime of program.
long starttime = System.nanoTime();
//How many primes found, how many primes wanted, loop counter.
int countOfPrimes = 0;
final int targetCountOfPrimes = 30;
long f = 1;
//Starting BigInteger at 1.
BigInteger biFactorial = BigInteger.ONE;
while (countOfPrimes < targetCountOfPrimes) {
//Each loop, multiply the number by the loop
//counter (f) to increase factorial much more quickly.
biFactorial = biFactorial.multiply(BigInteger.valueOf(f));
// one less than the factorial.
BigInteger biMinusOne = biFactorial.subtract(BigInteger.ONE);
// one more than the factorial.
BigInteger biPlusOne = biFactorial.add(BigInteger.ONE);

//Determine if the numbers are prime with a probability of 100
boolean primeMinus = biMinusOne.isProbablePrime(100);
boolean primePlus = biPlusOne.isProbablePrime(100);

//Make the big number look like a pretty string for output.
String biMinusOneString = convert(biMinusOne);
String biPlusOneString = convert(biPlusOne);

//If the number was prime, output and increment the primt counter.
if (primeMinus) {
countOfPrimes++;
System.out.println(
countOfPrimes + ": " + f + "! - 1 = " + biMinusOneString);
}
if (primePlus) {
countOfPrimes++;
System.out.println(countOfPrimes + ": " + f + "! + 1 = " + biPlusOneString);
}
//Increment loop counter.
f++;
}
//Calculate and display program runtime.
long stoptime = System.nanoTime();
long runtime = stoptime - starttime;
System.out.println("Program runtime: " + runtime + " ns (~" + runtime/1_000_000_000 + " seconds)");
}

//Method to make output pretty
private static String convert(BigInteger bi) {
String s = bi.toString();
int l = s.length();
String s2 = "";
if (l >= 40) {
s2 = s.substring(0, 19);
s2 += "..." + s.substring(s.length() - 20, s.length());
s2 += " : " + l + " digits";
} else {s2 = s;}
return s2;
}
}
```

### output

```1: 1! + 1 = 2
2: 2! + 1 = 3
3: 3! - 1 = 5
4: 3! + 1 = 7
5: 4! - 1 = 23
6: 6! - 1 = 719
7: 7! - 1 = 5039
8: 11! + 1 = 39916801
9: 12! - 1 = 479001599
10: 14! - 1 = 87178291199
11: 27! + 1 = 10888869450418352160768000001
12: 30! - 1 = 265252859812191058636308479999999
13: 32! - 1 = 263130836933693530167218012159999999
14: 33! - 1 = 8683317618811886495518194401279999999
15: 37! + 1 = 1376375309122634504...79581580902400000001 : 44 digits
16: 38! - 1 = 5230226174666011117...24100074291199999999 : 45 digits
17: 41! + 1 = 3345252661316380710...40751665152000000001 : 50 digits
18: 73! + 1 = 4470115461512684340...03680000000000000001 : 106 digits
19: 77! + 1 = 1451830920282858696...48000000000000000001 : 114 digits
20: 94! - 1 = 1087366156656743080...99999999999999999999 : 147 digits
21: 116! + 1 = 3393108684451898201...00000000000000000001 : 191 digits
22: 154! + 1 = 3089769613847350887...00000000000000000001 : 272 digits
23: 166! - 1 = 9003691705778437366...99999999999999999999 : 298 digits
24: 320! + 1 = 2116103347219252482...00000000000000000001 : 665 digits
25: 324! - 1 = 2288997460179102321...99999999999999999999 : 675 digits
26: 340! + 1 = 5100864472103711080...00000000000000000001 : 715 digits
27: 379! - 1 = 2484030746096470705...99999999999999999999 : 815 digits
28: 399! + 1 = 1600863071165597381...00000000000000000001 : 867 digits
29: 427! + 1 = 2906347176960734841...00000000000000000001 : 940 digits
30: 469! - 1 = 6771809666814951090...99999999999999999999 : 1051 digits
Program runtime: 6084297200 ns (~6 seconds)
```

## jq

The following jq program has been tested with both the C and Go implementations of jq. The latter supports unbounded-precision arithmetic, but the former has sufficient accuracy to compute the first 10 factorial primes. However, the implementation of `is_prime` used here is not sufficiently fast to compute more than the first 10 factorial primes in a reasonable amount of time.

```# The algorithm is quite fast because the state of `until` is just a number and we skip by 2 or 4
def is_prime:
. as \$n
| if (\$n < 2)         then false
elif (\$n % 2 == 0)  then \$n == 2
elif (\$n % 3 == 0)  then \$n == 3
else 5
| until( . <= 0;
if .*. > \$n then -1
elif (\$n % . == 0) then 0
else . + 2
|  if (\$n % . == 0) then 0
else . + 4
end
end)
| . == -1
end;

def factorial_primes:
foreach range(1; infinite) as \$i (1; . * \$i;
if ((.-1) | is_prime) then [(\$i|tostring) + "! - 1 = ", .-1] else empty end,
if ((.+1) | is_prime) then [(\$i|tostring) + "! + 1 = ", .+1] else empty end ) ;

limit(20; factorial_primes)
| .[1] |= (tostring | (if length > 40 then .[:20] + " .. " + .[-20:] else . end))

Invocation: jq -nr -f factorial-primes.jq

Output:
```1! + 1 = 2
2! + 1 = 3
3! - 1 = 5
3! + 1 = 7
4! - 1 = 23
6! - 1 = 719
7! - 1 = 5039
11! + 1 = 39916801
12! - 1 = 479001599
14! - 1 = 87178291199
# terminated
```

## Julia

```using Primes

limitedprint(n) = (s = string(n); n = length(s); return n <= 40 ? s : s[1:20] * "..." * s[end-19:end] * " (\$n digits)")

function showfactorialprimes(N)
for i in big"1":N
f = factorial(i)
isprime(f - 1) && println(lpad(i, 3), "! - 1 -> ", limitedprint(f - 1))
isprime(f + 1) && println(lpad(i, 3), "! + 1 -> ", limitedprint(f + 1))
end
end

showfactorialprimes(1000)
```
Output:
```  1! + 1 -> 2
2! + 1 -> 3
3! - 1 -> 5
3! + 1 -> 7
4! - 1 -> 23
6! - 1 -> 719
7! - 1 -> 5039
11! + 1 -> 39916801
12! - 1 -> 479001599
14! - 1 -> 87178291199
27! + 1 -> 10888869450418352160768000001
30! - 1 -> 265252859812191058636308479999999
32! - 1 -> 263130836933693530167218012159999999
33! - 1 -> 8683317618811886495518194401279999999
37! + 1 -> 13763753091226345046...79581580902400000001 (44 digits)
38! - 1 -> 52302261746660111176...24100074291199999999 (45 digits)
41! + 1 -> 33452526613163807108...40751665152000000001 (50 digits)
73! + 1 -> 44701154615126843408...03680000000000000001 (106 digits)
77! + 1 -> 14518309202828586963...48000000000000000001 (114 digits)
94! - 1 -> 10873661566567430802...99999999999999999999 (147 digits)
116! + 1 -> 33931086844518982011...00000000000000000001 (191 digits)
154! + 1 -> 30897696138473508879...00000000000000000001 (272 digits)
166! - 1 -> 90036917057784373664...99999999999999999999 (298 digits)
320! + 1 -> 21161033472192524829...00000000000000000001 (665 digits)
324! - 1 -> 22889974601791023211...99999999999999999999 (675 digits)
340! + 1 -> 51008644721037110809...00000000000000000001 (715 digits)
379! - 1 -> 24840307460964707050...99999999999999999999 (815 digits)
399! + 1 -> 16008630711655973815...00000000000000000001 (867 digits)
427! + 1 -> 29063471769607348411...00000000000000000001 (940 digits)
469! - 1 -> 67718096668149510900...99999999999999999999 (1051 digits)
546! - 1 -> 14130200926141832545...99999999999999999999 (1260 digits)
872! + 1 -> 19723152008295244962...00000000000000000001 (2188 digits)
974! - 1 -> 55847687633820181096...99999999999999999999 (2490 digits)
```

## LOLCODE

Basic task, based on the Algol 68 sample.

```OBTW find some factorial primes - primes that are f - 1 or f + 1
for some factorial f
TLDR

HAI 1.3

HOW IZ I TESTIN YR P    BTW PRIMALITY TEST WITH TRIAL DIVISHUN
DIFFRINT 3 AN SMALLR OF 3 AN P, O RLY?
YA RLY
FOUND YR BOTH SAEM P AN 2
MEBBE BOTH SAEM 0 AN MOD OF P AN 2
FOUND YR FAIL
NO WAI
I HAS A IZPRIME ITZ WIN
I HAS A N ITZ 3
I HAS A NSKWARED ITZ 9
IM IN YR PRIMELOOP UPPIN YR I TIL DIFFRINT NSKWARED AN SMALLR OF P AN NSKWARED
DIFFRINT 0 AN MOD OF P AN N, O RLY?
YA RLY
N R SUM OF N AN 2
NSKWARED R PRODUKT OF N AN N
NO WAI
IZPRIME R FAIL
NSKWARED R SUM OF P AN 1
OIC
IM OUTTA YR PRIMELOOP
FOUND YR IZPRIME
OIC
IF U SAY SO

HOW IZ I PADDIN YR FPNUMBR
I HAS A PAD ITZ ""
BOTH SAEM FPNUMBR AN SMALLR OF FPNUMBR AN 9, O RLY?
YA RLY
PAD R " "
OIC
FOUND YR SMOOSH PAD AN FPNUMBR MKAY
IF U SAY SO

HOW IZ I SHOWIN YR FPNUMBR AN YR N AN YR HOWDIFF AN YR FP
VISIBLE SMOOSH I IZ PADDIN YR FPNUMBR MKAY ...
AN ":: " AN I IZ PADDIN YR N MKAY ...
AN "! " AN HOWDIFF AN " 1 = " AN FP ...
MKAY
IF U SAY SO

I HAS A F ITZ 1
I HAS A N ITZ 0
I HAS A KOWNT ITZ 0
IM IN YR FPLOOP UPPIN YR I TIL BOTH SAEM KOWNT AN 10
N R SUM OF N AN 1
F R PRODUKT OF F AN N
I IZ TESTIN YR DIFF OF F AN 1 MKAY, O RLY?
YA RLY
KOWNT R SUM OF KOWNT AN 1
I IZ SHOWIN YR KOWNT AN YR N AN YR "-" AN YR DIFF OF F AN 1 MKAY
OIC
I IZ TESTIN YR SUM OF F AN 1 MKAY, O RLY?
YA RLY
KOWNT R SUM OF KOWNT AN 1
I IZ SHOWIN YR KOWNT AN YR N AN YR "+" AN YR SUM OF F AN 1 MKAY
OIC
IM OUTTA YR FPLOOP

KTHXBYE```
Output:
``` 1:  1! + 1 = 2
2:  2! + 1 = 3
3:  3! - 1 = 5
4:  3! + 1 = 7
5:  4! - 1 = 23
6:  6! - 1 = 719
7:  7! - 1 = 5039
8: 11! + 1 = 39916801
9: 12! - 1 = 479001599
10: 14! - 1 = 87178291199
```

## Lua

Translation of: ALGOL 68
```do -- find some factorial primes - primes that are f - 1 or f + 1
--      for some factorial f

function isPrime( p )
if p <= 1 or p % 2 == 0 then
return p == 2
else
local prime = true
local i     = 3
local rootP = math.floor( math.sqrt( p ) )
while i <= rootP and prime do
prime = p % i ~= 0
i     = i + 2
end
return prime
end
end

local f       = 1
local fpCount = 0
local n       = 0
local fpOp    = ""
while fpCount < 10 do
n    = n + 1
f    = f * n
fpOp = "-"
for fp = f - 1, f + 1, 2 do
if isPrime( fp ) then
fpCount = fpCount + 1
io.write( string.format( "%2d", fpCount  ), ":"
, string.format( "%4d", n        ), "! "
, fpOp, " 1 = ", fp, "\n"
)
end
fpOp = "+"
end
end

end
```
Output:
``` 1:   1! + 1 = 2
2:   2! + 1 = 3
3:   3! - 1 = 5
4:   3! + 1 = 7
5:   4! - 1 = 23
6:   6! - 1 = 719
7:   7! - 1 = 5039
8:  11! + 1 = 39916801
9:  12! - 1 = 479001599
10:  14! - 1 = 87178291199
```

## Mathematica/Wolfram Language

Translation of: Julia
```LimitedPrint[n_] := Module[{s = IntegerString[n], len},
len = StringLength[s];
If[len <= 40, s, StringJoin[StringTake[s, 20], "...", StringTake[s, -20], " (", ToString[len], " digits)"]]
]

ShowFactorialPrimes[N_] := Module[{f},
Do[
f = Factorial[i];
If[PrimeQ[f - 1], Print[IntegerString[i, 10, 3], "! - 1 -> ", LimitedPrint[f - 1]]];
If[PrimeQ[f + 1], Print[IntegerString[i, 10, 3], "! + 1 -> ", LimitedPrint[f + 1]]],
{i, 1, N}
]
]

ShowFactorialPrimes[1000]
```
Output:
```001! + 1 -> 2
002! + 1 -> 3
003! - 1 -> 5
003! + 1 -> 7
004! - 1 -> 23
006! - 1 -> 719
007! - 1 -> 5039
011! + 1 -> 39916801
012! - 1 -> 479001599
014! - 1 -> 87178291199
027! + 1 -> 10888869450418352160768000001
030! - 1 -> 265252859812191058636308479999999
032! - 1 -> 263130836933693530167218012159999999
033! - 1 -> 8683317618811886495518194401279999999
037! + 1 -> 13763753091226345046...79581580902400000001 (44 digits)
038! - 1 -> 52302261746660111176...24100074291199999999 (45 digits)
041! + 1 -> 33452526613163807108...40751665152000000001 (50 digits)
073! + 1 -> 44701154615126843408...03680000000000000001 (106 digits)
077! + 1 -> 14518309202828586963...48000000000000000001 (114 digits)
094! - 1 -> 10873661566567430802...99999999999999999999 (147 digits)
116! + 1 -> 33931086844518982011...00000000000000000001 (191 digits)
154! + 1 -> 30897696138473508879...00000000000000000001 (272 digits)
166! - 1 -> 90036917057784373664...99999999999999999999 (298 digits)
320! + 1 -> 21161033472192524829...00000000000000000001 (665 digits)
324! - 1 -> 22889974601791023211...99999999999999999999 (675 digits)
340! + 1 -> 51008644721037110809...00000000000000000001 (715 digits)
379! - 1 -> 24840307460964707050...99999999999999999999 (815 digits)
399! + 1 -> 16008630711655973815...00000000000000000001 (867 digits)
427! + 1 -> 29063471769607348411...00000000000000000001 (940 digits)
469! - 1 -> 67718096668149510900...99999999999999999999 (1051 digits)
546! - 1 -> 14130200926141832545...99999999999999999999 (1260 digits)
872! + 1 -> 19723152008295244962...00000000000000000001 (2188 digits)
974! - 1 -> 55847687633820181096...99999999999999999999 (2490 digits)

```

## Maxima

```block([i:1,count:0,result:[]],
while count<10 do (if primep(i!-1) or primep(i!+1) then (result:endcons(i,result),count:count+1),i:i+1),
result:map(lambda([x],[x,x!-1,x!+1]),result),
append(map(lambda([x],if primep(x[2]) then [x[1],x[2],"subtracted"]),result),map(lambda([x],if primep(x[3]) then [x[1],x[3],"added"]),result)),
unique(%%),
firstn(%%,10);
```
Output:
```[[1,2,"added"],[2,3,"added"],[3,5,"subtracted"],[3,7,"added"],[4,23,"subtracted"],[6,719,"subtracted"],[7,5039,"subtracted"],[11,39916801,"added"],[12,479001599,"subtracted"],[14,87178291199,"subtracted"]]
```

## Nim

Library: Nim-Integers

Nim standard integer types are limited to 64 bits. So we use an external library which provides arbitrary sized integers.

```import std/[math, strformat]

func isPrime(n: int): bool =
if n < 2: return false
if n == 2 or n == 3: return true
if n mod 2 == 0: return false
if n mod 3 == 0: return false
var d = 5
var step = 2
while d * d <= n:
if n mod d == 0:
return false
inc d, step
step = 6 - step
return true

echo "First 10 factorial primes:\n"
var count = 0
var n = 1
while count < 10:
let f = fac(n)
if isPrime(f - 1):
inc count
echo &"{count:>2}: {n:>3}! - 1 = {f - 1}"
if count < 10 and isPrime(f + 1):
inc count
echo &"{count:>2}: {n:>3}! + 1 = {f + 1}"
inc n

# Stretch.

import integers

func str(n: Integer): string =
## Return the string representation of an Integer.
result = \$n
if result.len > 40:
result = &"{result[0..19]}...{result[^20..^1]} ({result.len} digits)"

echo "\n\nNext 20 factorial primes:\n"
while count < 30:
let f: Integer = factorial(n)
if isPrime(f - 1):
inc count
echo &"{count:>2}: {n:>3}! - 1 = {str(f - 1)}"
if isPrime(f + 1):
inc count
echo &"{count:>2}: {n:>3}! - 1 = {str(f + 1)}"
inc n
```
Output:
```First 10 factorial primes:

1:   1! + 1 = 2
2:   2! + 1 = 3
3:   3! - 1 = 5
4:   3! + 1 = 7
5:   4! - 1 = 23
6:   6! - 1 = 719
7:   7! - 1 = 5039
8:  11! + 1 = 39916801
9:  12! - 1 = 479001599
10:  14! - 1 = 87178291199

Next 20 factorial primes:

11:  27! - 1 = 10888869450418352160768000001
12:  30! - 1 = 265252859812191058636308479999999
13:  32! - 1 = 263130836933693530167218012159999999
14:  33! - 1 = 8683317618811886495518194401279999999
15:  37! - 1 = 13763753091226345046...79581580902400000001 (44 digits)
16:  38! - 1 = 52302261746660111176...24100074291199999999 (45 digits)
17:  41! - 1 = 33452526613163807108...40751665152000000001 (50 digits)
18:  73! - 1 = 44701154615126843408...03680000000000000001 (106 digits)
19:  77! - 1 = 14518309202828586963...48000000000000000001 (114 digits)
20:  94! - 1 = 10873661566567430802...99999999999999999999 (147 digits)
21: 116! - 1 = 33931086844518982011...00000000000000000001 (191 digits)
22: 154! - 1 = 30897696138473508879...00000000000000000001 (272 digits)
23: 166! - 1 = 90036917057784373664...99999999999999999999 (298 digits)
24: 320! - 1 = 21161033472192524829...00000000000000000001 (665 digits)
25: 324! - 1 = 22889974601791023211...99999999999999999999 (675 digits)
26: 340! - 1 = 51008644721037110809...00000000000000000001 (715 digits)
27: 379! - 1 = 24840307460964707050...99999999999999999999 (815 digits)
28: 399! - 1 = 16008630711655973815...00000000000000000001 (867 digits)
29: 427! - 1 = 29063471769607348411...00000000000000000001 (940 digits)
30: 469! - 1 = 67718096668149510900...99999999999999999999 (1051 digits)
```

## OCaml

```let is_prime (_, n, _) =
let rec test x =
let d = n / x in x > d || x * d <> n && n mod (x + 2) <> 0 && test (x + 6)
in
if n < 5
then n lor 1 = 3
else n land 1 <> 0 && n mod 3 <> 0 && test 5

let factorials_plus_minus_one =
let rec next x y () =
Seq.Cons ((x, pred y, 0), Seq.cons (x, succ y, 1) (next (succ x) (succ x * y)))
in
next 1 1

let () =
let show (x, y, a) = Printf.printf "%3u! %c 1 = %u\n" x [|'-'; '+'|].(a) y in
factorials_plus_minus_one |> Seq.filter is_prime |> Seq.take 10 |> Seq.iter show
```
Output:
```  1! + 1 = 2
2! + 1 = 3
3! - 1 = 5
3! + 1 = 7
4! - 1 = 23
6! - 1 = 719
7! - 1 = 5039
11! + 1 = 39916801
12! - 1 = 479001599
14! - 1 = 87178291199
```

## Perl

Library: ntheory
```use v5.36;
use ntheory <is_prime factorial>;

sub show (\$d) { my \$l = length \$d; \$l < 41 ? \$d : substr(\$d,0,20) . '..' . substr(\$d,-20) . " (\$l digits)" }

my(\$cnt,\$n);
my \$fmt = "%2d: %3d! %s 1 = %s\n";

while () {
my \$f = factorial ++\$n;
if (is_prime \$f-1) { printf \$fmt, ++\$cnt, \$n, '-', show \$f-1 }
if (is_prime \$f+1) { printf \$fmt, ++\$cnt, \$n, '+', show \$f+1 }
last if \$cnt == 30;
}
```
Output:
``` 1:   1! + 1 = 2
2:   2! + 1 = 3
3:   3! - 1 = 5
4:   3! + 1 = 7
5:   4! - 1 = 23
6:   6! - 1 = 719
7:   7! - 1 = 5039
8:  11! + 1 = 39916801
9:  12! - 1 = 479001599
10:  14! - 1 = 87178291199
11:  27! + 1 = 10888869450418352160768000001
12:  30! - 1 = 265252859812191058636308479999999
13:  32! - 1 = 263130836933693530167218012159999999
14:  33! - 1 = 8683317618811886495518194401279999999
15:  37! + 1 = 13763753091226345046..79581580902400000001 (44 digits)
16:  38! - 1 = 52302261746660111176..24100074291199999999 (45 digits)
17:  41! + 1 = 33452526613163807108..40751665152000000001 (50 digits)
18:  73! + 1 = 44701154615126843408..03680000000000000001 (106 digits)
19:  77! + 1 = 14518309202828586963..48000000000000000001 (114 digits)
20:  94! - 1 = 10873661566567430802..99999999999999999999 (147 digits)
21: 116! + 1 = 33931086844518982011..00000000000000000001 (191 digits)
22: 154! + 1 = 30897696138473508879..00000000000000000001 (272 digits)
23: 166! - 1 = 90036917057784373664..99999999999999999999 (298 digits)
24: 320! + 1 = 21161033472192524829..00000000000000000001 (665 digits)
25: 324! - 1 = 22889974601791023211..99999999999999999999 (675 digits)
26: 340! + 1 = 51008644721037110809..00000000000000000001 (715 digits)
27: 379! - 1 = 24840307460964707050..99999999999999999999 (815 digits)
28: 399! + 1 = 16008630711655973815..00000000000000000001 (867 digits)
29: 427! + 1 = 29063471769607348411..00000000000000000001 (940 digits)
30: 469! - 1 = 67718096668149510900..99999999999999999999 (1051 digits)```

## Phix

```with javascript_semantics
include mpfr.e
atom tp = time(), tm = time()+16    -- per, max 16s runtime
mpz {e,f} = mpz_inits(2,1)
integer i = 1, c = 0
while time()<tm do
mpz_mul_si(f,f,i)
for k in {-1,+1} do
if mpz_prime(e) then
c += 1
string s = iff(k<0?"-":"+"),
es = mpz_get_short_str(e),
et = elapsed(time()-tp,0.1," (%s)")
printf(1,"%2d: %3d! %s %d = %s%s\n",{c,i,s,abs(k),es,et})
tp = time()
end if
end for
i += 1
end while
```
Output:
``` 1:   1! + 1 = 2
2:   2! + 1 = 3
3:   3! - 1 = 5
4:   3! + 1 = 7
5:   4! - 1 = 23
6:   6! - 1 = 719
7:   7! - 1 = 5039
8:  11! + 1 = 39916801
9:  12! - 1 = 479001599
10:  14! - 1 = 87178291199
11:  27! + 1 = 10888869450418352160768000001
12:  30! - 1 = 265252859812191058636308479999999
13:  32! - 1 = 263130836933693530167218012159999999
14:  33! - 1 = 8683317618811886495518194401279999999
15:  37! + 1 = 13763753091226345046315979581580902400000001
16:  38! - 1 = 523022617466601111760007224100074291199999999
17:  41! + 1 = 33452526613163807108170062053440751665152000000001
18:  73! + 1 = 44701154615126843408...03680000000000000001 (106 digits)
19:  77! + 1 = 14518309202828586963...48000000000000000001 (114 digits)
20:  94! - 1 = 10873661566567430802...99999999999999999999 (147 digits)
21: 116! + 1 = 33931086844518982011...00000000000000000001 (191 digits)
22: 154! + 1 = 30897696138473508879...00000000000000000001 (272 digits)
23: 166! - 1 = 90036917057784373664...99999999999999999999 (298 digits)
24: 320! + 1 = 21161033472192524829...00000000000000000001 (665 digits) (2.5s)
25: 324! - 1 = 22889974601791023211...99999999999999999999 (675 digits) (0.2s)
26: 340! + 1 = 51008644721037110809...00000000000000000001 (715 digits) (0.8s)
27: 379! - 1 = 24840307460964707050...99999999999999999999 (815 digits) (2.0s)
28: 399! + 1 = 16008630711655973815...00000000000000000001 (867 digits) (1.9s)
29: 427! + 1 = 29063471769607348411...00000000000000000001 (940 digits) (3.2s)
30: 469! - 1 = 67718096668149510900...99999999999999999999 (1,051 digits) (5.4s)
```

Items 15-17 are shown in full because that's still shorter than 20+length("...")+20+length(" (NN digits)").
Aside: Unfortunately the relative performance falls off a cliff under pwa/p2js by the 320! mark, and it'd probably need a few minutes to get to the 30th.

## Python

Library: gmpy2

This takes about 32 seconds to find the first 33 factorial primes on my machine (Ryzen 5 1500X).

```from itertools import count
from itertools import islice
from typing import Iterable
from typing import Tuple

import gmpy2

def factorials() -> Iterable[int]:
fact = 1
for i in count(1):
yield fact
fact *= i

def factorial_primes() -> Iterable[Tuple[int, int, str]]:
for n, fact in enumerate(factorials()):
if gmpy2.is_prime(fact - 1):
yield (n, fact - 1, "-")
if gmpy2.is_prime(fact + 1):
yield (n, fact + 1, "+")

def print_factorial_primes(limit=10) -> None:
print(f"First {limit} factorial primes.")
for n, fact_prime, op in islice(factorial_primes(), 1, limit + 1):
s = str(fact_prime)
if len(s) > 40:
s = f"{s[:20]}...{s[-20:]} ({len(s)} digits)"
print(f"{n}! {op} 1 = {s}")

if __name__ == "__main__":
import sys
print_factorial_primes(int(sys.argv[1]) if len(sys.argv) > 1 else 10)
```
Output:
```First 33 factorial primes.
1! + 1 = 2
2! + 1 = 3
3! - 1 = 5
3! + 1 = 7
4! - 1 = 23
6! - 1 = 719
7! - 1 = 5039
11! + 1 = 39916801
12! - 1 = 479001599
14! - 1 = 87178291199
27! + 1 = 10888869450418352160768000001
30! - 1 = 265252859812191058636308479999999
32! - 1 = 263130836933693530167218012159999999
33! - 1 = 8683317618811886495518194401279999999
37! + 1 = 13763753091226345046...79581580902400000001 (44 digits)
38! - 1 = 52302261746660111176...24100074291199999999 (45 digits)
41! + 1 = 33452526613163807108...40751665152000000001 (50 digits)
73! + 1 = 44701154615126843408...03680000000000000001 (106 digits)
77! + 1 = 14518309202828586963...48000000000000000001 (114 digits)
94! - 1 = 10873661566567430802...99999999999999999999 (147 digits)
116! + 1 = 33931086844518982011...00000000000000000001 (191 digits)
154! + 1 = 30897696138473508879...00000000000000000001 (272 digits)
166! - 1 = 90036917057784373664...99999999999999999999 (298 digits)
320! + 1 = 21161033472192524829...00000000000000000001 (665 digits)
324! - 1 = 22889974601791023211...99999999999999999999 (675 digits)
340! + 1 = 51008644721037110809...00000000000000000001 (715 digits)
379! - 1 = 24840307460964707050...99999999999999999999 (815 digits)
399! + 1 = 16008630711655973815...00000000000000000001 (867 digits)
427! + 1 = 29063471769607348411...00000000000000000001 (940 digits)
469! - 1 = 67718096668149510900...99999999999999999999 (1051 digits)
546! - 1 = 14130200926141832545...99999999999999999999 (1260 digits)
872! + 1 = 19723152008295244962...00000000000000000001 (2188 digits)
974! - 1 = 55847687633820181096...99999999999999999999 (2490 digits)
```

## Quackery

`!` is defined at Factorial#Quackery.

`isprime` is defined at Primality by trial division#Quackery.

```  [ dup 10 < if sp echo ]     is recho   ( n -->   )

[] 0
[ 1+ dup !
dup dip
[ 1 - isprime if
[ tuck negate join swap ] ]
1+ isprime if
[ tuck join swap ]
over size 9 > until ]
drop 10 split drop
witheach
[ i^ 1+
recho say ": "
dup abs tuck recho
0 < iff
[ say "! - 1 = " -1 ]
else
[ say "! + 1 = "  1 ]
swap ! + echo cr ]```
Output:
``` 1:  1! + 1 = 2
2:  2! + 1 = 3
3:  3! - 1 = 5
4:  3! + 1 = 7
5:  4! - 1 = 23
6:  6! - 1 = 719
7:  7! - 1 = 5039
8: 11! + 1 = 39916801
9: 12! - 1 = 479001599
10: 14! - 1 = 87178291199
```

## Racket

```#lang racket
(require gmp)

(define (factorial-boundary-stream)
(define (factorial-stream-iter n curr-fact)
(stream-cons `(- ,n ,(sub1 curr-fact))
(stream-cons `(+ ,n ,(add1 curr-fact))
(factorial-stream-iter (add1 n) (* curr-fact (+ n 1))))))
(factorial-stream-iter 1 1))

(define (format-large-number n)
(let* ([num-chars (number->string n)]
[num-len (string-length num-chars)])
(if (> num-len 40)
(string-append
(substring num-chars 0 19)
"..."
(substring num-chars (- num-len 19) num-len)
(format " (total ~a digits)" num-len))
n)))

(define (factorial-printer triple)
(let-values ([(op n fact) (apply values triple)])
(let ([fact (format-large-number fact)])
(displayln (format "~a! ~a 1 = ~a" n op fact)))))

(define (prime? n)
(not (zero? (mpz_probab_prime_p (mpz n) 10))))

(for ([i (in-stream
(stream-take
(stream-filter (λ (l) (prime? (third l))) (factorial-boundary-stream)) 30))]
[n (in-naturals 1)])
(begin
(display (format "~a:\t" n))
(factorial-printer i)))

;; time output of above code: 2.46 seconds
```
```1:	1! + 1 = 2
2:	2! + 1 = 3
3:	3! - 1 = 5
4:	3! + 1 = 7
5:	4! - 1 = 23
6:	6! - 1 = 719
7:	7! - 1 = 5039
8:	11! + 1 = 39916801
9:	12! - 1 = 479001599
10:	14! - 1 = 87178291199
11:	27! + 1 = 10888869450418352160768000001
12:	30! - 1 = 265252859812191058636308479999999
13:	32! - 1 = 263130836933693530167218012159999999
14:	33! - 1 = 8683317618811886495518194401279999999
15:	37! + 1 = 1376375309122634504...9581580902400000001 (total 44 digits)
16:	38! - 1 = 5230226174666011117...4100074291199999999 (total 45 digits)
17:	41! + 1 = 3345252661316380710...0751665152000000001 (total 50 digits)
18:	73! + 1 = 4470115461512684340...3680000000000000001 (total 106 digits)
19:	77! + 1 = 1451830920282858696...8000000000000000001 (total 114 digits)
20:	94! - 1 = 1087366156656743080...9999999999999999999 (total 147 digits)
21:	116! + 1 = 3393108684451898201...0000000000000000001 (total 191 digits)
22:	154! + 1 = 3089769613847350887...0000000000000000001 (total 272 digits)
23:	166! - 1 = 9003691705778437366...9999999999999999999 (total 298 digits)
24:	320! + 1 = 2116103347219252482...0000000000000000001 (total 665 digits)
25:	324! - 1 = 2288997460179102321...9999999999999999999 (total 675 digits)
26:	340! + 1 = 5100864472103711080...0000000000000000001 (total 715 digits)
27:	379! - 1 = 2484030746096470705...9999999999999999999 (total 815 digits)
28:	399! + 1 = 1600863071165597381...0000000000000000001 (total 867 digits)
29:	427! + 1 = 2906347176960734841...0000000000000000001 (total 940 digits)
30:	469! - 1 = 6771809666814951090...9999999999999999999 (total 1051 digits)
cpu time: 2440 real time: 2440 gc time: 3```

## Raku

```sub postfix:<!> (\$n) { constant @F = (1, 1, |[\*] 2..*); @F[\$n] }
sub abr (\$_) { .chars < 41 ?? \$_ !! .substr(0,20) ~ '..' ~ .substr(*-20) ~ " ({.chars} digits)" }

my \$limit;

for 1..* {
my \f = .!;
++\$limit and printf "%2d: %3d! - 1 = %s\n", \$limit, \$_, abr f -1 if (f -1).is-prime;
++\$limit and printf "%2d: %3d! + 1 = %s\n", \$limit, \$_, abr f +1 if (f +1).is-prime;
exit if \$limit >= 30
}
```
Output:
``` 1:   1! + 1 = 2
2:   2! + 1 = 3
3:   3! - 1 = 5
4:   3! + 1 = 7
5:   4! - 1 = 23
6:   6! - 1 = 719
7:   7! - 1 = 5039
8:  11! + 1 = 39916801
9:  12! - 1 = 479001599
10:  14! - 1 = 87178291199
11:  27! + 1 = 10888869450418352160768000001
12:  30! - 1 = 265252859812191058636308479999999
13:  32! - 1 = 263130836933693530167218012159999999
14:  33! - 1 = 8683317618811886495518194401279999999
15:  37! + 1 = 13763753091226345046..79581580902400000001 (44 digits)
16:  38! - 1 = 52302261746660111176..24100074291199999999 (45 digits)
17:  41! + 1 = 33452526613163807108..40751665152000000001 (50 digits)
18:  73! + 1 = 44701154615126843408..03680000000000000001 (106 digits)
19:  77! + 1 = 14518309202828586963..48000000000000000001 (114 digits)
20:  94! - 1 = 10873661566567430802..99999999999999999999 (147 digits)
21: 116! + 1 = 33931086844518982011..00000000000000000001 (191 digits)
22: 154! + 1 = 30897696138473508879..00000000000000000001 (272 digits)
23: 166! - 1 = 90036917057784373664..99999999999999999999 (298 digits)
24: 320! + 1 = 21161033472192524829..00000000000000000001 (665 digits)
25: 324! - 1 = 22889974601791023211..99999999999999999999 (675 digits)
26: 340! + 1 = 51008644721037110809..00000000000000000001 (715 digits)
27: 379! - 1 = 24840307460964707050..99999999999999999999 (815 digits)
28: 399! + 1 = 16008630711655973815..00000000000000000001 (867 digits)
29: 427! + 1 = 29063471769607348411..00000000000000000001 (940 digits)
30: 469! - 1 = 67718096668149510900..99999999999999999999 (1051 digits)```

## Ring

```see "working..." + nl

n = 0
num = 0
while true
n++
n1 = factorial(n) - 1
if isPrime(n1)
num++
see "" + num + ": " + n + "! - 1 = " + n1 + nl
ok
n2 = factorial(n) + 1
if isPrime(n2)
num++
see "" + num + ": " + n + "! + 1 = " + n2 + nl
ok
if num = 10
exit
ok
end
see "done..." + nl```
Output:
``` 1: 1! + 1 = 2
2: 2! + 1 = 3
3: 3! - 1 = 5
4: 3! + 1 = 7
5: 4! - 1 = 23
6: 6! - 1 = 719
7: 7! - 1 = 5039
8: 11! + 1 = 39916801
9: 12! - 1 = 479001599
10: 14! - 1 = 87178291199
```

## RPL

```≪ / LAST ROT * - #0 == ≫ 'MODNOT' STO

≪ R→B
IF DUP #3 ≤ THEN #2 / B→R
ELSE
IF DUP #1 AND #1 ≠ OVER #3 MODNOT OR
THEN DROP 0
ELSE
DUP B→R √ R→B → maxd
≪ #5 1 SF
WHILE 1 FS? OVER maxd ≤ AND REPEAT
IF DUP2 MODNOT THEN 1 CF END
#2 + IF DUP2 MODNOT THEN 1 CF END
#4 + END
≫
DROP2 1 FS?
END
END
≫ 'PRIM?' STO

≪
SWAP IP → d n
≪ n →STR "!" + d 0 > "+" "" IFTE +
d →STR + "=" + n FACT d + →STR +
≫ ≫ 'WRITE' STO

≪ { } 1
WHILE OVER SIZE 10 ≤ REPEAT
DUP FACT
IF DUP 1 - PRIM? THEN OVER -1 WRITE 4 ROLL SWAP + ROT ROT END
IF 1 + PRIM? THEN DUP 1 WRITE ROT SWAP + SWAP END
1 +
END
```
Output:
```1: { "1!+1=2" "2!+1=3" "3!-1=5" "3!+1=7" "4!-1=23" "6!-1=719" "7!-1=5039" "11!+1=39916801" "12!-1=479001599" "14!-1=87178291199" }
```

## Ruby

```require 'openssl'

factorial_primes = Enumerator.new do |y|
fact = 1
(1..).each do |i|
fact *= i
y << [i, "- 1", fact - 1] if OpenSSL::BN.new(fact - 1).prime?
y << [i, "+ 1", fact + 1] if OpenSSL::BN.new(fact + 1).prime?
end
end

factorial_primes.first(30).each do |a|
s = a.last.to_s
if s.size > 40 then
puts "%d! %s = " % a.first(2) + "#{s[0,20]}...#{s[-20,20]}"
else
puts "%d! %s = %d" % a
end
end
```
Output:
```1! + 1 = 2
2! + 1 = 3
3! - 1 = 5
3! + 1 = 7
4! - 1 = 23
6! - 1 = 719
7! - 1 = 5039
11! + 1 = 39916801
12! - 1 = 479001599
14! - 1 = 87178291199
27! + 1 = 10888869450418352160768000001
30! - 1 = 265252859812191058636308479999999
32! - 1 = 263130836933693530167218012159999999
33! - 1 = 8683317618811886495518194401279999999
37! + 1 = 13763753091226345046...79581580902400000001
38! - 1 = 52302261746660111176...24100074291199999999
41! + 1 = 33452526613163807108...40751665152000000001
73! + 1 = 44701154615126843408...03680000000000000001
77! + 1 = 14518309202828586963...48000000000000000001
94! - 1 = 10873661566567430802...99999999999999999999
116! + 1 = 33931086844518982011...00000000000000000001
154! + 1 = 30897696138473508879...00000000000000000001
166! - 1 = 90036917057784373664...99999999999999999999
320! + 1 = 21161033472192524829...00000000000000000001
324! - 1 = 22889974601791023211...99999999999999999999
340! + 1 = 51008644721037110809...00000000000000000001
379! - 1 = 24840307460964707050...99999999999999999999
399! + 1 = 16008630711655973815...00000000000000000001
427! + 1 = 29063471769607348411...00000000000000000001
469! - 1 = 67718096668149510900...99999999999999999999
```

## Sidef

```var factorial_primes = Enumerator({|f|
for k in (1..Inf) {
if (k!-1 -> is_prime) { f([k, -1]) }
if (k!+1 -> is_prime) { f([k, +1]) }
}
})

func abr(v) {
v.len <= 40 ? v : (v.to_s.first(20) + '..' + v.to_s.last(20) + " (#{v.len} digits)")
}

factorial_primes.first(30).each_2d {|k,i|
printf("%3d! %s %d = %s\n", k, (i.sgn < 0 ? '-' : '+'), i.abs, abr(k! + i))
}
```
Output:
```  1! + 1 = 2
2! + 1 = 3
3! - 1 = 5
3! + 1 = 7
4! - 1 = 23
6! - 1 = 719
7! - 1 = 5039
11! + 1 = 39916801
12! - 1 = 479001599
14! - 1 = 87178291199
27! + 1 = 10888869450418352160768000001
30! - 1 = 265252859812191058636308479999999
32! - 1 = 263130836933693530167218012159999999
33! - 1 = 8683317618811886495518194401279999999
37! + 1 = 13763753091226345046..79581580902400000001 (44 digits)
38! - 1 = 52302261746660111176..24100074291199999999 (45 digits)
41! + 1 = 33452526613163807108..40751665152000000001 (50 digits)
73! + 1 = 44701154615126843408..03680000000000000001 (106 digits)
77! + 1 = 14518309202828586963..48000000000000000001 (114 digits)
94! - 1 = 10873661566567430802..99999999999999999999 (147 digits)
116! + 1 = 33931086844518982011..00000000000000000001 (191 digits)
154! + 1 = 30897696138473508879..00000000000000000001 (272 digits)
166! - 1 = 90036917057784373664..99999999999999999999 (298 digits)
320! + 1 = 21161033472192524829..00000000000000000001 (665 digits)
324! - 1 = 22889974601791023211..99999999999999999999 (675 digits)
340! + 1 = 51008644721037110809..00000000000000000001 (715 digits)
379! - 1 = 24840307460964707050..99999999999999999999 (815 digits)
399! + 1 = 16008630711655973815..00000000000000000001 (867 digits)
427! + 1 = 29063471769607348411..00000000000000000001 (940 digits)
469! - 1 = 67718096668149510900..99999999999999999999 (1051 digits)```

## V (Vlang)

```import math

fn main() {
mut n, mut count := 0, 0
for count < 10 {
n++
f := math.factoriali(n)
if is_prime(f - 1) {
count++
println("\${count}: \${n}! - 1 = \${f - 1}")
}
if is_prime(f + 1) {
count++
println("\${count}: \${n}! + 1 = \${f + 1}")
}
}
}

fn is_prime(num i64) bool {
if num <= 1 {return false}
if num % 2 == 0 && num != 2 {return false}
for idx := 3; idx <= math.floor(num / 2) - 1; idx += 2 {
if num % idx == 0 {return false}
}
return true
}
```
Output:
``` 1: 1! + 1 = 2
2: 2! + 1 = 3
3: 3! - 1 = 5
4: 3! + 1 = 7
5: 4! - 1 = 23
6: 6! - 1 = 719
7: 7! - 1 = 5039
8: 11! + 1 = 39916801
9: 12! - 1 = 479001599
10: 14! - 1 = 87178291199
```

## Wren

### Basic

Library: Wren-math
Library: Wren-fmt
```import "./math" for Int
import "./fmt" for Fmt

System.print("First 10 factorial primes;")
var c = 0
var i = 1
var f = 1
while (true) {
for (gs in [[f-1, "-"], [f+1, "+"]]) {
if (Int.isPrime(gs[0])) {
Fmt.print("\$2d: \$2d! \$s 1 = \$d", c = c + 1, i, gs[1], gs[0])
if (c == 10) return
}
}
i = i + 1
f = f * i
}
```
Output:
```First 10 factorial primes;
1:  1! + 1 = 2
2:  2! + 1 = 3
3:  3! - 1 = 5
4:  3! + 1 = 7
5:  4! - 1 = 23
6:  6! - 1 = 719
7:  7! - 1 = 5039
8: 11! + 1 = 39916801
9: 12! - 1 = 479001599
10: 14! - 1 = 87178291199
```

### Stretch

Library: Wren-gmp

This takes about 28.5 seconds to reach the 33rd factorial prime on my machine (Core i7) with the last two being noticeably slower to emerge. Likely to be very slow after that as the next factorial prime is 1477! + 1.

```import "./gmp" for Mpz
import "./fmt" for Fmt

var limit = 33
var c = 0
var i = 1
var f = Mpz.one
System.print("First %(limit) factorial primes;")
while (true) {
f.mul(i)
var r = (i < 21) ? 1 : 0  // test for definite primeness below 2^64
for (gs in [[f-1, "-"], [f+1, "+"]]) {
if (gs[0].probPrime(15) > r) {
var s = gs[0].toString
var sc = s.count
var digs = sc > 40 ? "(%(sc) digits)" : ""
Fmt.print("\$2d: \$3d! \$s 1 = \$20a \$s", c = c + 1, i, gs[1], s, digs)
if (c == limit) return
}
}
i = i + 1
}
```
Output:
```First 33 factorial primes;
1:   1! + 1 = 2
2:   2! + 1 = 3
3:   3! - 1 = 5
4:   3! + 1 = 7
5:   4! - 1 = 23
6:   6! - 1 = 719
7:   7! - 1 = 5039
8:  11! + 1 = 39916801
9:  12! - 1 = 479001599
10:  14! - 1 = 87178291199
11:  27! + 1 = 10888869450418352160768000001
12:  30! - 1 = 265252859812191058636308479999999
13:  32! - 1 = 263130836933693530167218012159999999
14:  33! - 1 = 8683317618811886495518194401279999999
15:  37! + 1 = 13763753091226345046...79581580902400000001 (44 digits)
16:  38! - 1 = 52302261746660111176...24100074291199999999 (45 digits)
17:  41! + 1 = 33452526613163807108...40751665152000000001 (50 digits)
18:  73! + 1 = 44701154615126843408...03680000000000000001 (106 digits)
19:  77! + 1 = 14518309202828586963...48000000000000000001 (114 digits)
20:  94! - 1 = 10873661566567430802...99999999999999999999 (147 digits)
21: 116! + 1 = 33931086844518982011...00000000000000000001 (191 digits)
22: 154! + 1 = 30897696138473508879...00000000000000000001 (272 digits)
23: 166! - 1 = 90036917057784373664...99999999999999999999 (298 digits)
24: 320! + 1 = 21161033472192524829...00000000000000000001 (665 digits)
25: 324! - 1 = 22889974601791023211...99999999999999999999 (675 digits)
26: 340! + 1 = 51008644721037110809...00000000000000000001 (715 digits)
27: 379! - 1 = 24840307460964707050...99999999999999999999 (815 digits)
28: 399! + 1 = 16008630711655973815...00000000000000000001 (867 digits)
29: 427! + 1 = 29063471769607348411...00000000000000000001 (940 digits)
30: 469! - 1 = 67718096668149510900...99999999999999999999 (1051 digits)
31: 546! - 1 = 14130200926141832545...99999999999999999999 (1260 digits)
32: 872! + 1 = 19723152008295244962...00000000000000000001 (2188 digits)
33: 974! - 1 = 55847687633820181096...99999999999999999999 (2490 digits)
```

## XPL0

```func IsPrime(N);        \Return 'true' if N is prime
real N;  int  I;
[if N <= 2. then return N = 2.;
if Mod(N, 2.) = 0. then \even\ return false;
for I:= 3 to fix(sqrt(N)) do
[if Mod(N, float(I)) = 0. then return false;
I:= I+1;
];
return true;
];

func real Factorial(N); \Return N!
int N;  real F;
[F:= float(N);
while N > 1 do
[N:= N-1;
F:= F * float(N);
];
return F;
];

int  N, C;  real F;
[N:= 1;  C:= 0;
Format(1, 0);
repeat  F:= Factorial(N);
if IsPrime(F-1.) then
[IntOut(0, N);
Text(0, "! - 1 = ");
RlOut(0, F-1.);
CrLf(0);
C:= C+1;
];
if IsPrime(F+1.) then
[IntOut(0, N);
Text(0, "! + 1 = ");
RlOut(0, F+1.);
CrLf(0);
C:= C+1;
];
N:= N+1;
until   C >= 10;
]```
Output:
```1! + 1 = 2
2! + 1 = 3
3! - 1 = 5
3! + 1 = 7
4! - 1 = 23
6! - 1 = 719
7! - 1 = 5039
11! + 1 = 39916801
12! - 1 = 479001599
14! - 1 = 87178291199
```