The Smallest Nontrivial Solution to xk≡1 (modn) and Related Sequences

The smallest root of a polynomial congruence

“…x(log x) O (1) . On the other hand, a theorem of Tenenbaum concerning the count of numbers with large smooth components implies that the first sum on k is bounded, as x → ∞, by…”

“…(From here on in the argument, k 1 and k 2 denote generic X-smooth and X-rough numbers, respectively.) Discarding the condition that k 1 k 2 ∈ K and applying Cauchy-Schwarz, we see that 1) .…”

“…For the polynomials f (t) = (t + 2) n − 1 (with n ≥ 2), Theorem 5.1 was proved by Andrica and Crişan in [1]. It is easy to see that neither assumption on f in the statement of Theorem 5.1 can be removed.…”

“…x], we have (k) ≈ (log x) κ for a certain positive constant κ = κ f . This suggests the conjecture that κ is the "correct" value of c f in Theorem 1.2, in the sense that the smallest root of f modulo k is of size k/(log k) κ+o (1) as k → ∞ through a density 1 subset of R f .…”