Alateng Pan

Fibonacci Enumeration and Coverage of the Reverse Tree for the Odd-Increment Collatz Variant

We study the Collatz variant T: Z_{>0} -> Z_{>0} defined by T(n) = n/2 for even n and T(n) = n+1 for odd n. The reverse tree, obtained by iterating the inverse branches x -> 2x and x -> x-1 (the latter admissible when x is even) from the root 1, encodes the directed graph of the...

Fibonacci Enumeration and Coverage of the Reverse Tree for the Odd-Increment Collatz Variant

We study the Collatz variant T: Z_{>0} -> Z_{>0} defined by T(n) = n/2 for even n and T(n) = n+1 for odd n. The reverse tree, obtained by iterating the inverse branches x -> 2x and x -> x-1 (the latter admissible when x is even) from the root 1, encodes the directed graph of the...

Rotations About Body Points: Sharp Bounds for Rigid Motion

Let B ⊂ R^n be a compact rigid body with nonempty interior and diameter L, with n ≥ 2. We study the problem of realizing an orientation-preserving isometry of R^n as a composition of rotations whose centers lie on the current configuration of B. An earlier version of this work cl...

Rotations About Body Points: Sharp Bounds for Rigid Motion

Let B ⊂ R^n be a compact rigid body with nonempty interior and diameter L, with n ≥ 2. We study the problem of realizing an orientation-preserving isometry of R^n as a composition of rotations whose centers lie on the current configuration of B. An earlier version of this work cl...

Erdős–Graham Problem #411 – A Construction of Six Solution Families for the r=2 Case

We construct and verify six infinite families of solutions to the r=2 case of the Erdős–Graham iteration problem (Problem #411). Steinerberger [1] reduced the r=2 case to the equation φ(n) + φ(n+φ(n)) = n and proved that any solution must fall into one of two branches: either the...

Erdős–Graham Problem #411 – A Construction of Six Solution Families for the r=2 Case

We construct and verify six infinite families of solutions to the r=2 case of the Erdős–Graham iteration problem (Problem #411). Steinerberger [1] reduced the r=2 case to the equation φ(n) + φ(n+φ(n)) = n and proved that any solution must fall into one of two branches: either the...

Erdős–Graham Problem #411– A Complete Proof of the r=2 Case via the UF Framework

We present a complete proof of the r=2 case of the Erdős–Graham iteration problem (Problem #411). Steinerberger (2025) reduced this case to the equation phi(n) + phi(n + phi(n)) = n, finding six solution families and showing any additional solution must be of the form n = 2^a p^b...

The UF Operator – Definition, Construction, and Boundaries

This paper defines the UF operator (Unrestricted Function) as a new algebraic framework for implicit domain restriction. The operator allows functions to be written without explicit conditional statements by embedding domain restrictions into the algebraic structure via a formal...