 mersenneforum.org Double Cubic Frobenius Trinomial test
 Register FAQ Search Today's Posts Mark Forums Read 2022-10-10, 00:37 #1 paulunderwood   Sep 2002 Database er0rr 2·33·83 Posts Double Cubic Frobenius Trinomial test Let the test be: Code: {tst(n,a,b)=c=a+b;T=(x-a)*(x-b)*(x+c); trace(Mod(Mod(x,n),T-1)^n)==0&& trace(Mod(Mod(x,n),T+1)^n)==0} I have run n up to 10,000,000 with tst(n,1,1) with no output. I have also run: Code: for(n=4,10000000,if(!ispseudoprime(n)&&tst(n,1,2),print([n,factor(n)]))) [49, Mat([7, 2])] [343, Mat([7, 3])] [2401, Mat([7, 4])] [16807, Mat([7, 5])] [84035, [5, 1; 7, 5]] [98441, [7, 4; 41, 1]] [117649, Mat([7, 6])] [232897, [7, 4; 97, 1]] [823543, Mat([7, 7])] [5764801, Mat([7, 8])] Notice all passing numbers are divisible by 7. It's a fun test to play with!    2022-10-10, 08:31   #2
paulunderwood

Sep 2002
Database er0rr

2×33×83 Posts Quote:
 Originally Posted by paulunderwood Code: for(n=4,10000000,if(!ispseudoprime(n)&&tst(n,1,2),print([n,factor(n)]))) [49, Mat([7, 2])] [343, Mat([7, 3])] [2401, Mat([7, 4])] [16807, Mat([7, 5])] [84035, [5, 1; 7, 5]] [98441, [7, 4; 41, 1]] [117649, Mat([7, 6])] [232897, [7, 4; 97, 1]] [823543, Mat([7, 7])] [5764801, Mat([7, 8])] Notice all passing numbers are divisible by 7.
This 7 divisibility pattern is broken some time later with: [532758241, [97, 1; 673, 1; 8161, 1]]   2022-10-10, 13:41   #3
paulunderwood

Sep 2002
Database er0rr

2·33·83 Posts Quote:
 Originally Posted by paulunderwood Let the test be: Code: {tst(n,a,b)=c=a+b;T=(x-a)*(x-b)*(x+c); trace(Mod(Mod(x,n),T-1)^n)==0&& trace(Mod(Mod(x,n),T+1)^n)==0} I have run n up to 10,000,000 with tst(n,1,1) with no output.
Of interest to me are cases where a=3*A and b=3*B (A, B in N).

For example a=3*17 and b=3*17 produces numbers only divisible by D=17. But what is D for various A and B? Especially for A!=B?

Last fiddled with by paulunderwood on 2022-10-10 at 13:45   2022-10-10, 15:48   #4
paulunderwood

Sep 2002
Database er0rr

2·33·83 Posts Quote:
 Originally Posted by paulunderwood Of interest to me are cases where a=3*A and b=3*B (A, B in N). For example a=3*17 and b=3*17 produces numbers only divisible by D=17. But what is D for various A and B? Especially for A!=B?
Running over a list of Carmichael numbers (2^64) always gives a counterexample for various a and b, except this one which might just be a fluke:

Code:
for(v=1,#V,n=V[v];if(tst(n,16,16),print([n,factor(n)])))

Last fiddled with by paulunderwood on 2022-10-10 at 16:02  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post paulunderwood Miscellaneous Math 2 2021-10-14 22:03 ET_ Operazione Doppi Mersennes 1 2018-01-29 15:50 justinstevens42 Information & Answers 2 2018-01-22 16:03 ewmayer Computer Science & Computational Number Theory 5 2013-06-06 02:50 lidocorc Software 3 2008-12-03 15:12

All times are UTC. The time now is 17:30.

Sat Jan 28 17:30:36 UTC 2023 up 163 days, 14:59, 0 users, load averages: 1.15, 1.08, 1.07