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Old 2022-10-10, 00:37   #1
paulunderwood
 
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Cool 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!
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Old 2022-10-10, 08:31   #2
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Quote:
Originally Posted by paulunderwood View Post

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]]
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Old 2022-10-10, 13:41   #3
paulunderwood
 
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Quote:
Originally Posted by paulunderwood View Post
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
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Old 2022-10-10, 15:48   #4
paulunderwood
 
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Quote:
Originally Posted by paulunderwood View Post
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
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