1
$\begingroup$

Can you provide a proof for the following claim:

Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4}\right)^m\right)$ . Let $N= 4kp^n+1 $ such that $p$ is a prime number greater than $3$ , $k$ is an odd natural number , $3 \nmid k$ , $n\ge1$ and $4k <p^n$. Let $S_i=P_p(S_{i-1})$ with $S_0=P_{k}(10)$, then $N$ is a prime iff $S_{n} \equiv 0 \pmod{N}$ .

You can run this test here. I have verified this claim for $k \in [1,300]$ with $p \in [5,109]$ and $n \in [1,50]$ .

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ How did you come up with this conjecture? $\endgroup$
    – Milo Moses
    Commented Sep 8, 2020 at 4:30
  • 2
    $\begingroup$ @MiloMoses This was inspired by Lucas-Lehmer-Riesel test. $\endgroup$
    – Pedja
    Commented Sep 8, 2020 at 4:33
  • $\begingroup$ It is rather standard, for such tests, to prove that if $N$ is prime then it divides $S_n$, but the opposite direction looks difficult, and possibly even not true. Say, if $N$ is Carmichael number plus some other restrictions, it is a counterexample. $\endgroup$
    – Fedor Petrov
    Commented Sep 14 at 9:19

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .