Factorial primes

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Revision as of 18:11, 15 August 2022 by Wherrera (talk | contribs) (→‎{{header|Julia}}: formatting)
Factorial primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Definition

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

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.

Task

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
Related task



ALGOL 68

Basic task. Assumes LONG INT is at least 64 bits. <lang algol68>BEGIN # find some factorial primes - primes that are f - 1 or f + 1 # 

     #      for some factorial f                                        #
  1. 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;
  1. end of code based on the primality by trial divisio task #
   PROC show factorial prime = ( INT fp number, INT n, CHAR fp op, LONG INT fp )VOID:
      print( ( whole( fp number, -2 ), ":", whole( n, -4 )
             , "! ", fp op, " 1 = ", whole( fp, 0 )
             , newline
             )
           ); 
   LONG INT f        := 1;
   INT      fp count := 0;
   FOR n WHILE fp count < 10 DO
       f *:= n;
       IF  LONG INT fp = f - 1;
           is prime( fp )
       THEN
           show factorial prime( fp count +:= 1, n, "-", fp )
       FI;
       IF  LONG INT fp = f + 1;
           is prime( fp )
       THEN
           show factorial prime( fp count +:= 1, n, "+", fp )
       FI
   OD

END</lang>

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

F#

<lang fsharp> // 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]"}""") </lang>

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]

J

<lang 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</lang>

(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.)

Julia

<lang ruby>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)

</lang>

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. <lang lolcode>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
         NO WAI
            BOTH SAEM 0 AN MOD OF P AN 2, O RLY?
               YA RLY
                  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
     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 </lang>

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

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
        mpz_add_si(e,f,k)
        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)

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.

Raku

<lang perl6>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

}</lang>

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)

Wren

Basic

Library: Wren-math
Library: Wren-fmt

<lang ecmascript>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

}</lang>

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. <lang ecmascript>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

}</lang>

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)
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