20221218, 17:49  #34  
Sep 2002
Database er0rr
17×283 Posts 
Quote:
Now consider b=3 and b=3. There are corresponding pseudoprimes: Code:
{forstep(n=3,10000000,2, if(!ispseudoprime(n), forstep(b=3,3,6, if(gcd(b,n)==1&&Mod(b,n)^((n1)/2)==kronecker(b,n), z=znorder(Mod(b,n));if(z%2==0&&Mod(b,n)^(z/2)==1, for(r=1,z/2, D=lift(Mod(b,n)^(2*r)4*b);if(kronecker(D,n)==1&&Mod(Mod(x,n),x^2(Mod(b,n)^(2*r1)2)*x+1)^((n+1)/2)==kronecker(b,n), g=gcd(r1,n1);print([n,b,z,r,g]))))))));} [3281, 3, 16, 5, 4] [3281, 3, 16, 1, 3280] [432821, 3, 268, 68, 67] [432821, 3, 268, 1, 432820] [973241, 3, 232, 59, 58] [973241, 3, 232, 1, 973240] [1551941, 3, 508, 128, 127] [1551941, 3, 508, 1, 1551940] [2202257, 3, 112, 29, 28] [2202257, 3, 112, 1, 2202256] [2545181, 3, 460, 116, 115] [2545181, 3, 460, 1, 2545180] [3020093, 3, 268, 68, 67] [3020093, 3, 268, 1, 3020092] [3028133, 3, 268, 68, 67] [3028133, 3, 268, 1, 3028132] [4561481, 3, 616, 155, 154] [4561481, 3, 616, 1, 4561480] [4923521, 3, 640, 161, 160] [4923521, 3, 640, 1, 4923520] [5173601, 3, 928, 233, 232] [5173601, 3, 928, 1, 5173600] [5193161, 3, 280, 71, 70] [5193161, 3, 280, 1, 5193160] [5774801, 3, 400, 101, 100] [5774801, 3, 400, 1, 5774800] [6710177, 3, 352, 89, 88] [6710177, 3, 352, 1, 6710176] [9846401, 3, 640, 161, 160] [9846401, 3, 640, 1, 9846400] Last fiddled with by paulunderwood on 20221218 at 18:03 

20221218, 19:09  #35 
Dec 2022
2×3×7×13 Posts 
As this clearly isn't an attempt at a proof, I'm going to ignore you like everyone else does.

20221219, 00:06  #36  
Sep 2002
Database er0rr
11313_{8} Posts 
Quote:
You mentioned earlier we have Fermat PRP and strong Fermat. Both these are no foolproof and PSPs (pseudoprimes) can be constructed. e.g. Carmichael (absolute PSP) numbers for Fermat PRP. Lucas sequence can be used as an alternative. It is now known whether absolute PSPs exist for Lucas. Then there is BPSW. It is believed that counterexamples exist for this test: See: https://www.d.umn.edu/~jgreene/baillie/BailliePSW.html and http://pseudoprime.com/dopo.pdf I am not convinced by my own reasoning  albeit shallow  that my tests for b=3 and b=12 are true or not. Until I have a proof I will keep shtum! 

20221219, 00:40  #37  
If I May
"Chris Halsall"
Sep 2002
Barbados
10110010011100_{2} Posts 
Quote:
I greatly enjoy watching people do things. Sometimes things work; sometimes things don't... Please keep working on this until you have either proven everyone else incorrect, or you have proven to yourself that you are incorrect. I go down *so* many rabbit holes that are a waste of R&D time. But, rarely, there gold down in those holes! 8^) P.S. Has anyone worked with the PRUs on some ARM kit? Kinda cool... 

20221219, 14:31  #38 
Romulan Interpreter
"name field"
Jun 2011
Thailand
10358_{10} Posts 

20221220, 04:35  #39 
Sep 2002
Database er0rr
11313_{8} Posts 
The case for b=2
I'm back by popular demand!
The key with x^23^r*x3 was to transform it to z^2((3)^(2*r1)2)*z+1 and note that (3)^(2*r1)2 = 3*(3^(2*(r1))+1)+32. So we want to "avoid" 3^(4*(r1))1. Thus avoiding cyclotomic z^2z+1 With the transformation of x^22^r*x+1 to z^2(2^(2*r1)2)*z+1 the "key" is to notice that (2)^(2*r1)  2 = (2)^(2*r1)1+12. where we want to "avoid" 2^(2*(2*r1))1. So we take gcd(2*r1,n)==1. We avoid cyclotomic z^2+z+1 Here is some more code I am running. The patterns are (multiplicative order) z is odd for pseudoprimes and these only exist for n%6==5. Using Feitsma's 2PSP list: Code:
{b=2;for(v=1,#V, n=V[v];if(Mod(b,n)^((n1)/2)==kronecker(b,n), z=znorder(Mod(b,n));for(r=1,z, a=lift(Mod(b,n)^r);D=lift(a^24*b);if(kronecker(D,n)==1&&Mod(Mod(x,n),x^2a*x+b)^(n+1)==b, g=gcd(2*r1,n1);print([n,n%6,b,z,g,r])))));} Here is a table of patterns for pseudoprimes: Code:
b==2; z odd; kronecker(2,n)==1; n%6==5; requires gcd(2*r1,n1); exists a pseudoprime for r=(z+1)/2; b==3; z even; kronecker(3,n)==1; n%12==5; requires gcd(r1,n1); exists a pseudoprime for r=z/4+1; Last fiddled with by paulunderwood on 20221220 at 06:54 
20221222, 14:36  #40  
Sep 2002
Database er0rr
1001011001011_{2} Posts 
Quote:
Code:
[452295831401, 5, 2, 9716, 2429, 3644] [452295831401, 5, 2, 9716, 2429, 8502] Another pattern might be z < sqrt(n). Last fiddled with by paulunderwood on 20221222 at 14:38 

20230130, 00:05  #41 
Sep 2002
Database er0rr
1001011001011_{2} Posts 
The test x^(n+1)==3 (mod n, x^23^r*x3) and gcd(r1,n1) pans out for n < 10^13 and all r.
The verification with GMP+primesieve took several weeks on a dual core Celeron. If the code was run on some big iron machines then a much higher bound could be achieved, but I believe not high enough to find a pseudoprime. 
20230517, 15:32  #42 
Sep 2002
Database er0rr
17×283 Posts 
More on 12
If instead I compute over x^2x(12)^r I observe the following:
On the other hand there is only output of pseudoprimes over x^2x+12^r for r==z where z is the order of 12 Last fiddled with by paulunderwood on 20230517 at 15:33 
20230629, 21:25  #43 
Sep 2002
Database er0rr
17×283 Posts 
mart_t and I checked for pseudoprimes w.r.t. x^212^r*x12 for n<10^12 some months back na drew a blank. I have continued checking but only Carmichael numbers and have reached 10^13 (on a 1.1GHz Celeron core with parigp).

20230707, 22:06  #44 
Sep 2002
Database er0rr
12CB_{16} Posts 
Having searched Wiki for "12" I found that: "Twelve is the smallest weight for which a cusp form exists... SL(2,Z)....."
https://en.wikipedia.org/wiki/12_(number) Being a bear of little brain I ask the question: "Could this have something to do with the GCDless test based on x^212^r*x12?" Last fiddled with by paulunderwood on 20230707 at 23:22 
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