Conjectures About Cyclic Numbers: Resolutions and Counterexamples

We settle 22 conjectures of Cohen about cyclic numbers (positive integers n with (n, φ (n) ) =1), proving 16 and disproving 6, and we completely resolve a related OEIS problem about sequences whose running averages are Fibonacci numbers. Highlights include: asymptotics for cyclics between consecutive squares with a second-order term (Conj. ~9), Legendre- and k-fold Oppermann-type results in short quadratic intervals (Conj. ~6, Conj. ~20, and twin cyclics between cubes, Conj. ~32), gap and growth analogs (Visser, Rosser, Ishikawa, and a sum-3-versus-sum-2 inequality; Conj. ~47, ~52, ~54, ~56), limiting ratios (Vrba and Hassani; Conj. ~60, ~61), and structure results for Sophie Germain cyclics (Conj. ~36, ~37). We also resolve two Firoozbakht-type conjectures for cyclics (Conj. ~41--42). On the negative side we exhibit counterexamples to the Panaitopol, Dusart, and Carneiro analogs (Conj. ~59, ~53, ~50--51). Finally, for the lexicographically least sequence of pairwise distinct positive integers whose running averages are Fibonacci numbers (A248982), we give explicit closed forms for all n and prove Fried's Conjecture~2 asserting the disjointness of the parity-defined value sets (equivalently, F₍+₂+2nF₍+₁ is never a Fibonacci number). Proofs in this paper were assisted by GPT-5.