Belphegor Primes
October 31, 2014
We use the primality tester from a previous exercise and write this function to calculate Belphegor numbers:
(define (belphegor x)
(let ((n (+ (expt 10 (+ x x 4))
(* 666 (expt 10 (+ x 1)))
1)))
n))
Then we can list the Belphegor primes, though be aware that the Belphegor numbers quickly grow large, so the calculation is quite slow:
> (do ((n 0 (+ n 1))) (#f)
(when (prime? (belphegor n))
(display n) (newline)))
0
13
42
506
...
You can run the program at http://programmingpraxis.codepad.org/5hBfAjlj. Happy Halloween!
Programming Praxis 
Java, with probability chek:
public static void main(String[] args) {
int n = -1;
while (true) {
n++;
if (new BigInteger(“1” + StringUtils.repeat(“0”, n) + “666” + StringUtils.repeat(“0”, n) + “1”).isProbablePrime(10)) {
System.out.println(“n: ” + n + ” is probably (1 – 1/2^” + 10 + “) a prime.”);
}
}
}
In J, though I’m still a rookie:
ip=:1=#@q: NB. is prime
bn=:1+(10x^+&1)*666+10^+&3 NB. nth Belphegor number
bp=:ip@bn#] NB. ns yielding prime Belphegor numbers
NB. for example:
pb 11 12 13 NB. prints 13
pb i.14 NB. prints 0 13
NB. alternatively, all at once:
BP=:(1=#@q:)@(1+(10x^+&1)*666+10^+&3)#]
J formatting via code blocks leaves something to be desired; apologies!
from itertools import count from your_math_library import isprime # Use gmpy if it's available. try: from gmpy import mpz; tens = mpz(1) except: tens = 1 for n in count(): belph = (tens * 1000 + 666) * tens * 10 + 1 if isprime(belph): print n, "prime" elif n%100 == 0: print n tens *= 10Perl, with progress indicators every 100. About a minute to get to 2623 so faster than gmpy or Pari/GP for me. It’ll be a long>/i> time getting to 28291. These numbers make easy BLS75-T5 proofs so the proof doesn’t add too much time. Replacing with is_prob_prime (BPSW) or is_prime (BPSW+one or more random base M-R tests) makes it run slightly faster, but between 2623 and 28291 they’d all be the same regardless.
use ntheory qw/is_prime/; for (0 .. 100000) { say if is_provable_prime("1"."0"x$_."666"."0"x$_."1"); say " - $_" unless $_ % 100; }One likely, and relatively easy way to exocise Belphegore, is by having the possessed person work nonstop. Belphegore is the demon of sloth and laziness. He will likely find a new host at some point in this process.