Bounded Orbits of Quadratic Collatz-type Recursions

The periodic integral orbits of polynomial recursions with integer coefficients

We show that polynomial recursions $x_{n+1}=x_{n}^{m}-k$ where $k,m$ are integers and $m$ is positive have no nontrivial periodic integral orbits for $m\geq3$. If $m=2$ then the recursion has integral two-cycles for infinitely many values of $k$ but no higher period orbits. We also show that these statements are true for all quadratic recursions.

通过有限型移位和ζ函数研究二次易系数猜想

The Quadratic Easy Coefficients Conjecture via Finite-Type Shifts and Zeta Functions

本文通过有限型移位和ζ函数，证明了二次易系数猜想，推导了易系数公式，并论证了权重递推的非奇异性与唯一反向延拓性。