20170222, 23:24  #1 
"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 theproduct by thatproduct(+) 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 20170222 at 23:36 
20170222, 23:55  #2 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2·3^{2}·509 Posts 
Lovely title!

20170222, 23:57  #3 
"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.

20170223, 00:00  #4  
"Forget I exist"
Jul 2009
Dumbassville
8369_{10} Posts 
Quote:
etc. Last fiddled with by science_man_88 on 20170223 at 00:04 

20170223, 03:54  #5 
"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("\nBMT100AAlternativeFactorials=FalsorialsRandomFactors.gp\n") allocatemem() allocatemem() allocatemem() allocatemem() allocatemem() allocatemem() n=12345679001 isprime(n) for (i=3,19,{ falsorial=i; while(falsorial<10^10, falsorial=falsorial*(falsorial1); theGcd=gcd(falsorial,n); if(theGcd!=1,print("*** Found a factor: ",theGcd);next(19);); ); }) print("**** End of Run ****") 
20170223, 04:23  #6  
"Sam"
Nov 2016
319_{10} Posts 
Quote:


20170223, 04:28  #7  
"Rashid Naimi"
Oct 2015
Remote to Here/There
2·7·139 Posts 
Quote:


20170223, 05:01  #8 
"Rashid Naimi"
Oct 2015
Remote to Here/There
79A_{16} Posts 
https://factordb.com/index.php?id=1100000000905790309
Code:
print("\nBMT100CAlternativeFactorials=FalsorialsRandomFactors.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*(falsorial1);\\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 ****") 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*(falsorial1);\\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 20170223 at 05:14 
20170223, 14:25  #9 
"Forget I exist"
Jul 2009
Dumbassville
8369_{10} 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 
20170223, 17:17  #10 
"Mark"
Apr 2003
Between here and the
1784_{16} Posts 
Sounds like someone needs to do some prime hunting (and not me this time).

20170223, 18:31  #11  
"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.
From Carpetpool's link: Quote:

