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-   -   x!^n+/-1 or x#^n+/-1? (https://www.mersenneforum.org/showthread.php?t=26187)

 rogue 2020-11-13 16:01

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.

 Dr Sardonicus 2020-11-13 16:39

[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

$$\Phi_{n}(x!)\text{, }\Phi_{2n}(x!)\text{,}$$

or the same polynomials with the argument x! replaced with x#?

 rogue 2020-11-13 17:55

[QUOTE=Dr Sardonicus;563108]Presumably you mean

$$\Phi_{n}(x!)\text{, }\Phi_{2n}(x!)\text{,}$$

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].

 axn 2020-11-13 18:02

[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.

 R. Gerbicz 2020-11-13 18:44

[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].

 rogue 2020-11-13 19:40

As for algebraic factors, I had forgotten about those.

 Dylan14 2020-11-13 20:16

Doing a quick search with pfgw with the -f1 flag, the following numbers are prime or 3-PRP 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!^2-1[/code]

 rogue 2020-11-13 20:59

[QUOTE=Dylan14;563138]Doing a quick search with pfgw with the -f1 flag, the following numbers are prime or 3-PRP 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!^2-1[/code][/QUOTE]

I wouldn't expect anything else on the -1 side as x^n-1 always factors as (x-1) 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.

 a1call 2020-11-14 11:51

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 non-generalized factorial primes there are plenty of twin-primes in the generated form which are highly unreserved.

 Dr Sardonicus 2020-11-14 13:58

[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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