I am trying to prove why a polynomial with coefficients of the sequences of Collatz starting by prime number $n$ to the first appearance of $1$ is an irreducible polynomial (I found composite $n$ where the polynomial is not irreducible), also I tried even the reversed polynomial.
> here is my Pari/GP code
collatz(n)={
local(t=n);local(seq= [n]);
while(t!=1,
if(t%2==0,
seq = concat(seq,t/2); t=t/2,
seq = concat(seq,3*t+1);t=3*t+1;
)
);
return(seq);
}
{forprime(p=2,1000000,
C = collatz(p);
P = Pol(C);
Q = Polrev(C);
if (polisirreducible(P)*polisirreducible(Q)==0 , print(p)
))
}
So I can conjecture the following
Assuming the conjecture of Collatz is True, the polynomial P is irreducible for any prime.
That sounds cool. So you have checked for all primes less than a million? I presume though that some of those polynomials would have very large degree, much larger than a million - how long does it take you to test irreducibility?
Thanks James Tuite ,
it took less than 30 minutes on my laptop.
and the maximum degree was 431
Ah, I see, so the degree of the polynomial is the length of the Collatz sequence. I'm with you :) Thanks
I just had a go with some small values and agree with you for those. So I'll call it Rahmani's Conjecture :) I'd guess that there is a large counterexample, but if it is true it would be amazing. And impossible to prove for now sadly.
I completed the proof:
https://www.researchgate.net/publication/395507038_Mirror-Modular_Spine_Congruence_Saturation_and_Covariant_CRT_Closure_Solve_the_3x_1_Puzzle
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