Following conjecture on an infinite set of numbers satisfying the PSW-conjecture might be of academic interest in the understanding thereof.
Would you have any pointers on how to prove or disprove conjecture P?
Conjecture P. The odd p-th member with prime $p\geq11$ of a Lucas sequence $U_{p}(P,Q)$ with positive nonsquare discriminant $\Delta=P^2-4Q$ is prime if and only if $2^{U_{p}-1} \equiv 1 \pmod{U_{p}}$.
The Lucas sequence is defined by the recurrence $U_0=0$, $U_1=1$ and $U_{n+1}=PU_n-QU_{n-1}$.
Following broader Conjecture C was verified for all such numbers below $10^{30}$ representing a sample of 32'058'023 primes and 808'916'009 composites.
Conjecture C. The odd n-th member with $n\geq11$ of a Lucas sequence $U_{n}(P,Q)$ with positive nonsquare discriminant $\Delta=P^2-4Q$ is never pseudoprime to base 2 and if prime its order always greater than $\sqrt{U_{n}}$.
Finally a narrower Conjecture O suggesting the order always includes one of the primitive factors of its neighbor was verified for all such numbers below $10^{100}$ representing a sample of 351'847'849 primes.
Conjecture O. The odd p-th member with prime $p \geq 11$ of a Lucas sequence $U_{p}(P,-1)$ is prime if \begin{eqnarray*} \epsilon = \pm1 &\equiv & p \pmod{4} \\ 2^\frac{U_{p}-1}{U_{p+\epsilon}^{\star}} &\not\equiv & 1 \pmod{U_{p}} \\ 2^{U_{p}-1} & \equiv & 1 \pmod{U_{p}} \end{eqnarray*} where $U^{\star}$ denotes the primitive part only.
Useful articles in this context include
Baillie, Robert, and Samuel S. Wagstaff. "Lucas pseudoprimes" Mathematics of Computation 35, no. 152 (1980): 1391-1417.
Pomerance, Carl. "Are there counter-examples to the Baillie - PSW primality test?", 1984
Pomerance, Carl, John L. Selfridge, and Samuel S. Wagstaff. "The Pseudoprimes to $25 \cdot 10^9$" Mathematics of Computation 35, no. 151 (1980): 1003-1026.
Rosen, Michael I. "A proof of the Lucas-Lehmer test" American Mathematical Monthly 95, no. 9 (1988): 855-856.
Yabuta, Minoru. "A simple proof of Carmichael's theorem on primitive divisors" Fibonacci Quarterly 39, no. 5 (2001): 439-443.
