Fulsorials

[a1call]

Hi all,
There are Factorials, primorials, multfactorials.
But as far as I know the following is not coined.
I would like to introduce Fulsorials to you.
You can calculate Fulsorials by:
Multiplying 2 consecutive integers,
Then multiplying the-product by that-product(+-) 1
And continue indefinitely.
Every new multiplication will be by a new coprime and No primality test is required.
It could be used for finding random large factors to prime candidates without having to prove those factors primes.
As an example of Fulsorials:
6$=2*3*7*43*1807*3263443
It can also be useful for finding large PRPs.

[Batalov]

Lovely title!

[a1call]

Somehow I knew you'd like it. I gave the title more thought than the subject.

[science_man_88]

Originally Posted by a1call

Somehow I knew you'd like it. I gave the title more thought than the subject.

not really because there are alternatives https://en.wikipedia.org/wiki/Fallin...ing_factorials allows two types of factorials for example. edit: and there's https://en.wikipedia.org/wiki/Gamma_function as an extension.

etc.

[a1call]

Here is a hopefully useful code for finding random factors (have not tested it myself yet, but expect a decent performance). Tweak the for and while loop parameters to suit your needs.
Also would appreciate large integers posted here for trial runs.

Thank you in advance.

print("\nBMT-100-A-Alternative-Factorials=Falsorials-Random-Factors.gp\n")

allocatemem()
allocatemem()
allocatemem()
allocatemem()
allocatemem()
allocatemem()

n=12345679001
isprime(n)
for (i=3,19,{
  falsorial=i;
  while(falsorial<10^10,
    falsorial=falsorial*(falsorial-1);
    theGcd=gcd(falsorial,n);
    if(theGcd!=1,print("*** Found a factor: ",theGcd);next(19););
  );
})
print("**** End of Run ****")

[carpetpool]

Originally Posted by a1call

Hi all,
There are Factorials, primorials, multfactorials.
But as far as I know the following is not coined.
I would like to introduce Fulsorials to you.
You can calculate Fulsorials by:
Multiplying 2 consecutive integers,
Then multiplying the-product by that-product(+-) 1
And continue indefinitely.
Every new multiplication will be by a new coprime and No primality test is required.
It could be used for finding random large factors to prime candidates without having to prove those factors primes.
As an example of Fulsorials:
6$=2*3*7*43*1807*3263443
It can also be useful for finding large PRPs.

A specific type of "Fulsorials" are Sylvester's Sequence. You have a much more general idea of this.

[a1call]

Originally Posted by carpetpool

A specific type of "Fulsorials" are Sylvester's Sequence. You have a much more general idea of this.

Thank you for that carpetpool. I am only 137 years too late.

[a1call]

https://factordb.com/index.php?id=1100000000905790309

print("\nBMT-100-C-Alternative-Factorials=Falsorials-Random-Factors.gp\n")

allocatemem()
allocatemem()
allocatemem()
allocatemem()
allocatemem()
allocatemem()

n= 4883945163367692991
isprime(n)
for (i=3,19^4,{
  falsorial=i;
  while(falsorial<10^100000,
    falsorial=falsorial*(falsorial-1);\\print(falsorial);
    theGcd=gcd(falsorial,n);
    if(theGcd!=1,print("*** Found a factor: ",theGcd);next(19););
    theGcd=gcd(falsorial+1,n);
    if(theGcd!=1,print("*** Found a factor: ",theGcd);next(19););
  );
})
print("**** End of Run ****")

ETA:
https://factordb.com/index.php?id=1100000000905788578

 n= 254035168468567119979994968319537
%2 = 254035168468567119979994968319537
(00:10) gp > isprime(n)
%3 = 0
(00:10) gp > for (i=3,19^4,{
  falsorial=i;
  while(falsorial<10^100000,
    falsorial=falsorial*(falsorial-1);\\print(falsorial);
    theGcd=gcd(falsorial,n);
    if(theGcd!=1,print("*** Found a factor: ",theGcd);next(19););
    theGcd=gcd(falsorial+1,n);
    if(theGcd!=1,print("*** Found a factor: ",theGcd);next(19););
  );
})
*** Found a factor: 41
(00:10) gp > print("**** End of Run ****")
**** End of Run ****

[science_man_88]

for others that may be interested you have a lot of alternatives:

https://en.wikipedia.org/wiki/Factorial

talks of hyperfactorials and superfactorials

https://en.wikipedia.org/wiki/Alternating_factorial is another one

and the bottom links on some of these include:

https://en.wikipedia.org/wiki/Bhargava_factorial and
https://en.wikipedia.org/wiki/Exponential_factorial

[rogue]

Sounds like someone needs to do some prime hunting (and not me this time).

[a1call]

I am not sure who you are referring to and I can't speak for Mr Sylvester. But I wouldn't have a clue how to fully factor any of the larger terms. So if anyone feels any off this is of any use you have my blessings to use them.
From Carpetpool's link:

The recurrence by which it is defined allows the numbers in the sequence to be factored more easily than other numbers of the same magnitude, but, due to the rapid growth of the sequence, complete prime factorizations are known only for a few of its members.

And my factoring code seems to miss a lot of larger prime factors. But I still think it can be useful for large PRPs.