Factorial primes: Difference between revisions
→{{header|Wren}}: More efficient basic version. |
→{{header|ALGOL 68}}: Tweak again |
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=={{header|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 #
#
# is prime PROC based on the one in the primality by trial division task #
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prime
FI;
# end of code
PROC show factorial prime = ( INT fp number, INT n, CHAR fp op, LONG INT fp )VOID:
Line 51:
);
LONG INT f := 1;
INT
FOR n WHILE fp count < 10 DO
f *:= n;
IF LONG INT fp
THEN
show factorial prime( fp count +:= 1
FI;
IF LONG INT fp
THEN
show factorial prime( fp count +:= 1
FI
OD
|
Revision as of 15:06, 13 August 2022
- 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 #
- 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 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
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.)
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
<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) {
if (Int.isPrime(f-1)) { Fmt.print("$2d: $2d! - 1 = $d", c = c + 1, i, f - 1) if (c == 10) return } if (Int.isPrime(f+1)) { Fmt.print("$2d: $2d! + 1 = $d", c = c + 1, i, f + 1) 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
This takes about 28.5 seconds to reach the 33rd factorial prime on my machine (Core i7) with the last two being noticably 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 if ((f-1).probPrime(15) > r) { var s = (f-1).toString var sc = s.count var digs = sc > 40 ? "(%(sc) digits)" : "" Fmt.print("$2d: $3d! - 1 = $20a $s", c = c + 1, i, s, digs) if (c == limit) return } if ((f+1).probPrime(15) > r) { var s = (f+1).toString var sc = s.count var digs = sc > 40 ? "(%(sc) digits)" : "" Fmt.print("$2d: $3d! + 1 = $20a $s", c = c + 1, i, 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)