20201113, 16:01  #1 
"Mark"
Apr 2003
Between here and the
2×3^{2}×347 Posts 
x!^n+/1 or x#^n+/1?
Is there any interest from anyone here in searching for primes in the form x!^n+/1 or x#^n+/1? I would refer to them as "power factorial" or "power primorial". As far as I know these forms have never been searched or n>1. It would be fairly easy to modify fsieve and psieve to support them.

20201113, 16:39  #2  
Feb 2017
Nowhere
3×1,447 Posts 
Quote:
or the same polynomials with the argument x! replaced with x#? 

20201113, 17:55  #3  
"Mark"
Apr 2003
Between here and the
2·3^{2}·347 Posts 
Quote:
5! = 5*4*3*2*1 so 5!^3 = (5*4*3*2*1)^3 5# = 5*3*2 so 5#^3 = (5*3*2)^3 Here is a list of factorial primes: https://primes.utm.edu/top20/page.php?id=30 Here is a list of primorial primes: https://primes.utm.edu/top20/page.php?id=5. 

20201113, 18:02  #4 
Jun 2003
2^{3}×607 Posts 

20201113, 18:44  #5  
"Robert Gerbicz"
Oct 2005
Hungary
2645_{8} Posts 
Quote:
Code:
? factor(polcyclo(20,2)) %24 = [ 5 1] [41 1] Last fiddled with by R. Gerbicz on 20201113 at 18:46 Reason: grammar 

20201113, 19:40  #6 
"Mark"
Apr 2003
Between here and the
2·3^{2}·347 Posts 
As for algebraic factors, I had forgotten about those.

20201113, 20:16  #7 
"Dylan"
Mar 2017
2×281 Posts 
Doing a quick search with pfgw with the f1 flag, the following numbers are prime or 3PRP for the form x!^2 +/ 1 up to x = 2000:
Code:
0!^2+1 1!^2+1 2!^2+1 3!^2+1 4!^2+1 5!^2+1 9!^2+1 10!^2+1 11!^2+1 13!^2+1 24!^2+1 65!^2+1 76!^2+1 2!^21 
20201113, 20:59  #8  
"Mark"
Apr 2003
Between here and the
2×3^{2}×347 Posts 
Quote:
So from those perspectives, these forms are not that interesting. 

20201114, 11:51  #9 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{6}×31 Posts 
Remove the +/1 and add +/k and divide by k where k  x!^n
Things will get interesting Ali candidates will be proveable via N/+1 method since they will be of form where N has prime factors equal to or less than x. Additionally if both (x!^n+/k)/k are prime then they are twin primes. Here is non twin, n=1 example: https://www.mersenneforum.org/showpo...8&postcount=19 ETA Unlike nongeneralized factorial primes there are plenty of twinprimes in the generated form which are highly unreserved. Last fiddled with by a1call on 20201114 at 12:28 
20201114, 13:58  #10  
Feb 2017
Nowhere
3×1,447 Posts 
Quote:
There is at most one "intrinsic" prime factor. Let P be the evaluation of the cyclotomic polynomial, and n the exponent. If P has an intrinsic prime factor, it is gcd(P,n). 
