The least self-power root of a prime

For a prime q let f(q) = min{n ≥ 2 : q ∣ nn − 1} be its least self-power root. Writing V(X) for the number of its values in [2, X], we prove an explicit inequality giving V(X) ≥ X1/2−ε and, from the Baker–Harman theorem on shifted primes, V(X) ≥ X0.677−o(1). We explain why the fixed-point estimates of Kurlberg, Luca and Shparlinski, sharpened by Felix and Kurlberg, imply that Crocker's irreducible primes have relative density zero. Among primes q ≡ 1 (mod 3), the elements of order 3, and those of order 6, have a representative divisible by their order with relative density 1/3 each. We show that f(q) ≥ q1/2−ε for almost all primes, classify the automatic cofactors, exhibit coprime self-order pairs, and classify the small cases. We conjecture that f is surjective onto {n ≥ 2}, verified for n ≤ 104. 2020 Mathematics Subject Classification: Primary 11A07; Secondary 11N37, 11R45, 11Y11. The accompanying evidence archive contains the finite computations and Lean proof sources; the LaTeX source archive is also supplied.