Let r, s be coprime, with r odd and s0, and put z=r+s-2, D=r²-2s², Q=r²+2s², =z², = z², Uₙ=ⁿ-ⁿ-, Bₙ=ⁿ+ⁿ2. Assume that f3 is odd and Bf= Aᶠ with \1\ and A>0. Fix a prime power pᵉ f, put h=f/pᵉ, and assume h>1 and the clean alternative p A. Then p Bₕ, |Bₕ|=Rᶠ, and R^pᵉ1, an explicit correction factor Cd, a squarefree defect Ed\1, P^+ (d) \, and an integer Sd>0 satisfy Cd|M₂₃U|=EdSdᶠ. On the support-saturated sublattice (h) d, the correction disappears and, with =P^+ (h), |M₂₃U|=^dSdᶠ, d\0, 1\, where the external kernels Sd are pairwise coprime, satisfy (Sd, 2fQ) =1, and the defect positions form at most one complete -power chain. Applying the theorem of Bilu--Hanrot--Voutier at every saturated d>15 yields pairwise distinct external primes qd fQ with zU (qd) =2d, 4d qd--2qd, qdᶠ Uₐ₃--₂ₐ₃. Their product gives an explicit lower bound for Q. Exponent-supported lower primes simultaneously carry a near-depth-f Lucas--Wieferich condition and a deep ordinary Q-Wieferich condition. These are necessary structural conditions and do not prove the Beal conjecture.
Ueoka et al. (Wed,) studied this question.