Conjectures About Cyclic Numbers: Resolutions and Counterexamples

Abstract

We settle 22 conjectures of Cohen about cyclic numbers (positive integers $n$ with $\gcd(n,φ(n))=1$), 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 (\seqnum{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, $F_{n+2}+2nF_{n+1}$ is never a Fibonacci number). Proofs in this paper were assisted by GPT-5.