Perfect-Power Quadratic-Trace Lucas Towers

Abstract

Let $r,s\in\Z$ be coprime, with $r$ odd and $s\ne0$, and put $z=r+s\sqrt{-2},\qquad D=r^2-2s^2,\qquad Q=r^2+2s^2,$ $B_n=\frac{z^{2n}+\bar z^{2n}}2.$ Assume that, for an odd integer $f\ge3$, one has a perfect-power endpoint $B_f=\sigma A^f$ with $\sigma\in\{\pm1\}$ and $A>0$. For each $p^e\Vert f$ we construct a normalized prime-power tower and prove an exact coprimality splitting of its layers. If $p\mid A$, the tower forces the depth-$e$ Wieferich condition $p^{e+1}\mid Q^{p-1}-1.$ If $p\nmid A$, every prime $q$ in the lower layer carries a residue packet modulo the full square modulus $q^{2v_q(B_{f/p^e})}$. Its Teichm\"uller component satisfies an exact phase relation; a nontrivial packet implies $p^e\mid q-1,\qquad$ $v_p(\ord_q(Q))=v_p(q-1),\qquad$ $v_p(\ord_q(p))=v_p(q-1)-e+1.$ These conditions define a weighted acyclic support graph and yield a terminal classification at the largest prime divisor of $f$. The unit lower-layer alternative collapses to the pure prime-power boundary by Mih\u{a}ilescu's theorem. Using the primitive-divisor theorem of Bilu--Hanrot--Voutier, we also obtain a distinguished lower-support prime of rank $2h$ whenever the reduced index $h$ exceeds $15$. The results are necessary structural conditions; they do not prove the Beal conjecture.