20200205, 12:27  #1 
"AMD YES!"
Jan 2020
Bellevue, WA
40_{10} Posts 
Conjectured K question
How do you calculate the conjectured K for each base?

20200205, 20:27  #2 
Nov 2016
2^{2}×3×7×23 Posts 
Using this program in PARI: (using c(n) to find the conjectured smallest Sierpinski number base n)
Code:
is(n)=forprime(p=2,50000,if(n%p==0,return(0)));1 a(n,k,b)=n*b^k+1 b(n,r)=for(k=1,5000,if(gcd(n+1,r1)>1  is(a(n,k,r)),return(0)));1 c(n)=for(k=1,5000000,if(b(k,n),return(k))) Code:
is(n)=forprime(p=2,50000,if(n%p==0,return(0)));1 a(n,k,b)=n*b^k1 b(n,r)=for(k=1,5000,if(gcd(n1,r1)>1  is(a(n,k,r)),return(0)));1 c(n)=for(k=1,5000000,if(b(k,n),return(k))) 
20200205, 22:51  #3 
Jan 2017
2·13 Posts 
Sweety's code appears to be bruteforcing the CK search. Doing it rigorously requires advanced mathematics (modulo arithmetic, orders mod n, Chinese Remainder Theorem, sometimes quadratic, cubic residue analysis)
A basic outline of the steps it takes to find a CK in the easiest case (2cover): 1) Check if there are two primes with order 2 (modulo the base) 2) Calculate b^1 and b^2 (modulo each prime) 3) Calculate the 2 possible CRT solutions (b^1 mod p1, b^2 mod p2) and (b^2 mod p1, b^1 mod p2). Whichever CRT solution is smaller is the conjectured k (unless a smaller k is found with a higherorder covering set, as is the case with many bases where the order2 primes are 3 and some large prime in the hundreds or thousands) There are CK's found by this project that are FAR more complex than this, just look at base 3 with a 36cover. CK's more complex than 2 cover are best calculated using software such as Robert Gerbicz's bigcovering.exe or the Sweety PARI code below, as the number of possible covering sets and thus number of CRT solutions that need to be checked grows exponentially for each higher power. Last fiddled with by NHoodMath on 20200205 at 22:53 Reason: considering bases with 2covering sets but which have CK's with higher order coversets 
20200206, 00:41  #4 
May 2007
Kansas; USA
7·1,447 Posts 
There is a program out there called covering.exe or bigcovering.exe and there was a thread here at CRUS that talked about it. I don't have time to try to find them but with a diligent effort they can be found. You cannot brute force huge conjectures. The covering programs use covering set logic to only search k's that could possibly be the conjectures.
If anyone can post a link to the covering or bigcovering programs that would be helpful. Last fiddled with by gd_barnes on 20200206 at 00:43 
20200206, 01:50  #5  
"Dylan"
Mar 2017
2^{2}·3^{2}·13 Posts 
Quote:


20200206, 02:47  #6 
"AMD YES!"
Jan 2020
Bellevue, WA
28_{16} Posts 
I'm in China....
No google, no discord, no youtube, blahblahblah 
20200206, 03:17  #7  
Jan 2017
32_{8} Posts 
Quote:


20200206, 03:41  #8 
"AMD YES!"
Jan 2020
Bellevue, WA
2^{3}·5 Posts 
Though I still can't access Google... 
20200206, 03:53  #9 
"Curtis"
Feb 2005
Riverside, CA
3^{3}×5×31 Posts 

20200206, 05:24  #10 
Romulan Interpreter
Jun 2011
Thailand
20532_{8} Posts 
Or just skip it... (we have plenty of work to do)
(this reminds us of the guy who didn't want to go to year 2000, he wanted to stay in 1999, or something like that... haha, he didn't know that the new century actually started in 2001) 
20200206, 09:31  #11 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT)
13034_{8} Posts 

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