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.

[QUOTE=rogue;563104]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.[/QUOTE]Presumably you mean
[tex]\Phi_{n}(x!)\text{, }\Phi_{2n}(x!)\text{,}[/tex] or the same polynomials with the argument x! replaced with x#? 
[QUOTE=Dr Sardonicus;563108]Presumably you mean
[tex]\Phi_{n}(x!)\text{, }\Phi_{2n}(x!)\text{,}[/tex] or the same polynomials with the argument x! replaced with x#?[/QUOTE] I'm not fully understanding your notation. 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: [url]https://primes.utm.edu/top20/page.php?id=30[/url] Here is a list of primorial primes: [url]https://primes.utm.edu/top20/page.php?id=5[/url]. 
[QUOTE=rogue;563121]I'm not fully understanding your notation.[/QUOTE]
x^n+/1 has trivial algebraic factors. So he was suggesting to look at the nth cyclotomic polynomial instead  i.e. what is left after removing the algebraic factors. 
[QUOTE=axn;563122]x^n+/1 has trivial algebraic factors. So he was suggesting to look at the nth cyclotomic polynomial instead  i.e. what is left after removing the algebraic factors.[/QUOTE]
Exactly. There is a not hard pattern for these factors: if additionally you remove all lower exponents factors then for the remaining p prime factors what divides polcyclo(n,x) we have p%n=1. You really need to do this, because for example: [CODE] ? factor(polcyclo(20,2)) %24 = [ 5 1] [41 1] [/CODE] Ofcourse after the sieve with these special primes you need to do only a few gcd's on the remaining candidates. [or start the sieve removing these]. 
As for algebraic factors, I had forgotten about those.

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[/code] 
[QUOTE=Dylan14;563138]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[/code][/QUOTE] I wouldn't expect anything else on the 1 side as x^n1 always factors as (x1) as a factor. Note that on the +1 side any primes must also be GFNs which means that n will always be a power of 2 for primes. So from those perspectives, these forms are not that interesting. 
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: [url]https://www.mersenneforum.org/showpost.php?p=546848&postcount=19[/url] ETA Unlike nongeneralized factorial primes there are plenty of twinprimes in the generated form which are highly unreserved. 
[QUOTE=R. Gerbicz;563126]Exactly. There is a not hard pattern for these factors: if additionally you remove all lower exponents factors then for the remaining p prime factors what divides polcyclo(n,x) we have p%n=1. You really need to do this, because for example:
[CODE] ? factor(polcyclo(20,2)) %24 = [ 5 1] [41 1] [/CODE] Ofcourse after the sieve with these special primes you need to do only a few gcd's on the remaining candidates. [or start the sieve removing these].[/QUOTE] Right, the factor 5 in the above is sometimes called an "intrinsic" prime factor. It shows up because 5 divides polcyclo(4,2). An additional factor of 5 shows up in polcyclo(4*5, 2); another in polcyclo(4*5^2, 2), another in polcyclo(4*5^3, 2) and so on. This is described in detail in [url=https://www.maths.lancs.ac.uk/~jameson/cyp.pdf]The Cyclotomic Polynomials[/url]. 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). 
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