Least m such that n! + m is prime

Least m such that n! + m is prime 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.

Find the minimum positive integer m such that n factorial plus m is prime.

E.G.

   0! = 1. The next prime greater than 1 is 2. 2 - 1 = 1, so a(0) = 1.
   1! = 1. The next prime greater than 1 is 2. 2 - 1 = 1, so a(1) = 1.
   2! = 2. The next prime greater than 2 is 3. 3 - 2 = 1, so a(2) = 1.
   3! = 6. The next prime greater than 6 is 7. 7 - 6 = 1, so a(3) = 1.
   4! = 24. The next prime greater than 24 is 31. 31 - 24 = 5, so a(4) = 5.

and so on...

Task

• Find and display the first fifty terms in the series. (0! through 49!)
• Find and display the position and value of the first m greater than 1000.

Stretch

• Find and display the position and value of each the first m greater than 2000, 3000, 4000 ... 10,000.

See also

• OEIS:A033932 - Least positive m such that n! + m is prime

Raku

my @f = lazy flat 1, [\×] 1..*;

sink @f[700]; # pre-reify for concurrency

my @least-m = lazy (^∞).hyper(:2batch).map: {(1..*).first: -> \n {(@f[$_] + n).is-prime}};

say "Least positive m such that n! + m is prime; first fifty:\n" 
 ~ @least-m[^50].batch(10)».fmt("%3d").join: "\n";

for (1..10).map: * × 1e3 {
    my $key = @least-m.first: * > $_, :k;
    printf "\nFirst m > $_ is %d at position %d\n", @least-m[$key], $key;
}

Output:

Least positive m such that n! + m is prime; first fifty:
  1   1   1   1   5   7   7  11  23  17
 11   1  29  67  19  43  23  31  37  89
 29  31  31  97 131  41  59   1  67 223
107 127  79  37  97  61 131   1  43  97
 53   1  97  71  47 239 101 233  53  83

First m > 1000 is 1069 at position 107

First m > 2000 is 3391 at position 192

First m > 3000 is 3391 at position 192

First m > 4000 is 4943 at position 284

First m > 5000 is 5233 at position 384

First m > 6000 is 6131 at position 388

First m > 7000 is 9067 at position 445

First m > 8000 is 9067 at position 445

First m > 9000 is 9067 at position 445

First m > 10000 is 12619 at position 599

Wren

Library: Wren-gmp

Library: Wren-fmt

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

var fact = Mpz.one
var p = Mpz.new()
var diffs = List.filled(50, 0)
var n = 0
var t = 1000
var limit = 10000
while (true) {
    if (n > 0) fact.mul(n)
    p.nextPrime(fact)
    var m = p.sub(fact).toNum
    if (n < 50) diffs[n] = m
    if (n == 49) {
        System.print("Least positive m such that n! + m is prime; first 50:")
        Fmt.tprint("$3d ", diffs, 10)
        System.print()
    } else if (m > t) {
        while (true) {
            Fmt.print("First m > $,6d is $,6d at position $d", t, m, n)
            t = t + 1000
            if (m <= t) break
        }
        if (t > limit) return
    }
    n = n + 1
}

Output:

Least positive m such that n! + m is prime; first 50:
  1    1    1    1    5    7    7   11   23   17 
 11    1   29   67   19   43   23   31   37   89 
 29   31   31   97  131   41   59    1   67  223 
107  127   79   37   97   61  131    1   43   97 
 53    1   97   71   47  239  101  233   53   83 

First m >  1,000 is  1,069 at position 107
First m >  2,000 is  3,391 at position 192
First m >  3,000 is  3,391 at position 192
First m >  4,000 is  4,943 at position 284
First m >  5,000 is  5,233 at position 384
First m >  6,000 is  6,131 at position 388
First m >  7,000 is  9,067 at position 445
First m >  8,000 is  9,067 at position 445
First m >  9,000 is  9,067 at position 445
First m > 10,000 is 12,619 at position 599
First m > 11,000 is 12,619 at position 599
First m > 12,000 is 12,619 at position 599