The least self-power root of a prime

For a prime q let f(q) = min{n ≥ 2 : q | n^n - 1} be its least self-power root (for odd q, Crocker's smallest fundamental number exceeding 1). Writing V(X) for the number of values of f in [2, X], we prove an explicit inequality giving V(X) ≥ X^(1/2 - ε) and, from the Baker-Harman theorem on shifted primes, V(X) ≥ X^(0.677 - o(1)). For the primes with f(q) = q - 1 (Crocker's irreducible primes) we prove I(Y) ≤ (c + o(1)) Y/log Y with c = 47279/302400 = 0.1563..., from Chebotarev's theorem in a compositum of five fields and the equidistribution of the roots of a fixed quadratic polynomial to prime moduli in progressions; the companion paper lowers the constant to 887399/6652800 = 0.1333... by an equidistribution theorem in ray classes of Q(ζ_3). 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) ≥ q^(1/2 - ε) for all but o(π(Q)) primes q ≤ Q, and determine the cofactors k for which (q - 1)/k is a self-power root of every prime q ≡ 1 (mod k), q > k + 1: exactly k = 1 and k = 4. We show that coprime x, y ≥ 2 can share a prime q with ord_q(x) = x and ord_q(y) = y, exhibiting four such pairs, and classify the small cases. We conjecture that f is surjective onto {n ≥ 2}, verified for n ≤ 10^4. 2020 Mathematics Subject Classification: Primary 11A07; Secondary 11N37, 11R45, 11Y11. The evidence record (ZIP, 87 files) contains the census, distribution, irreducible-prime, quadratic-root, certificate, pair and interval calculations and the Lean 4 proof sources listed in SHA256SUMS.txt.