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Old 2017-02-22, 23:24   #1
a1call
 
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"Rashid Naimi"
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Default Fulsorials

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.

Last fiddled with by a1call on 2017-02-22 at 23:36
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Old 2017-02-22, 23:55   #2
Batalov
 
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Lovely title!
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Old 2017-02-22, 23:57   #3
a1call
 
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Somehow I knew you'd like it. I gave the title more thought than the subject.
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Old 2017-02-23, 00:00   #4
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Quote:
Originally Posted by a1call View Post
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.

Last fiddled with by science_man_88 on 2017-02-23 at 00:04
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Old 2017-02-23, 03:54   #5
a1call
 
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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.

Code:
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 ****")
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Old 2017-02-23, 04:23   #6
carpetpool
 
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Quote:
Originally Posted by a1call View Post
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.
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Old 2017-02-23, 04:28   #7
a1call
 
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Quote:
Originally Posted by carpetpool View Post
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.
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Old 2017-02-23, 05:01   #8
a1call
 
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https://factordb.com/index.php?id=1100000000905790309

Code:
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

Code:
 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 ****

Last fiddled with by a1call on 2017-02-23 at 05:14
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Old 2017-02-23, 14:25   #9
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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
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Old 2017-02-23, 17:17   #10
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Sounds like someone needs to do some prime hunting (and not me this time).
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Old 2017-02-23, 18:31   #11
a1call
 
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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:

Quote:
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.
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