This manuscript develops a new “killer prime” obstruction to primitive solutions of the generalized Fermat (Beal) equation aˣ+bʸ=cᶻ, with x, y, z≥3. Working in the p-th cyclotomic field, we attach to each rational slope a finite–field obstruction polynomial Φp (t), identified with a hypergeometric trace function for a rigid rank-2 local system. Katz’s equidistribution theorems imply that for any fixed slope there is a density-one set of primes p for which Φp (t) ≠0, forcing a λ-adic valuation mismatch and ruling out global solutions for the corresponding exponent patterns.
Chris Chenault (Mon,) studied this question.