Existence of Wall–Sun–Sun primes

Determine whether any Wall–Sun–Sun primes exist, equivalently whether there is a prime \(p>5\) satisfying the relevant Lucas-number congruence \(L_p\equiv1\pmod{p^2}\).

Background

In dimension one, a1(n)=Lna_1(n)=L_n, the nn-th Lucas number. For primes p>5p>5, the condition νp(a1(p)−1)≥2\nu_p(a_1(p)-1)\ge2 is identified with the classical Wall–Sun–Sun, or Fibonacci–Wieferich-type, condition.

The paper notes that extensive computational searches have found no such prime within very large ranges. This unresolved number-theoretic question is relevant because the paper uses the failure or existence of this condition to analyze tameness of primes.

References

So every prime that has been checked is tame for $d=1$, and whether Wall-Sun-Sun primes exist at all remains an open conjecture, and we do not resolve it here.

— The Algebraicity Problem for Hard-Core Entropy Constants on the Discrete Hypertori  (2608.28332 - Svoray, 28 Aug 2026) in Remark 2.??, Section 2, Remark \ref{rem:WSS}