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 2017-02-22, 23:24 #1 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 2·7·139 Posts 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  2017-02-22, 23:55 #2 Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2·32·509 Posts Lovely title!  2017-02-22, 23:57 #3 a1call "Rashid Naimi" Oct 2015 Remote to Here/There 2×7×139 Posts Somehow I knew you'd like it. I gave the title more thought than the subject. 2017-02-23, 00:00 #4 science_man_88 "Forget I exist" Jul 2009 Dumbassville 836910 Posts Quote:  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. Last fiddled with by science_man_88 on 2017-02-23 at 00:04  2017-02-23, 03:54 #5 a1call "Rashid Naimi" Oct 2015 Remote to Here/There 2×7×139 Posts 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 ****") 2017-02-23, 04:23 #6 carpetpool "Sam" Nov 2016 31910 Posts Quote:  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.

2017-02-23, 04:28   #7
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

2·7·139 Posts

Quote:
 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.

 2017-02-23, 05:01 #8 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 79A16 Posts 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
 2017-02-23, 14:25 #9 science_man_88     "Forget I exist" Jul 2009 Dumbassville 836910 Posts 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
 2017-02-23, 17:17 #10 rogue     "Mark" Apr 2003 Between here and the 178416 Posts Sounds like someone needs to do some prime hunting (and not me this time).
2017-02-23, 18:31   #11
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

2·7·139 Posts

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.

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.