Double Cubic Frobenius Trinomial test

[paulunderwood]

Let the test be:

{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:

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!

[paulunderwood]

Originally Posted by paulunderwood

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]]

[paulunderwood]

Originally Posted by paulunderwood

Let the test be:

{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?

[paulunderwood]

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:

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