bibkey: bradshaw2025collatz authors: Z. P. Bradshaw year: 2025 title: “On a Family of Solutions to Arithmetic Differential Equations Involving the Collatz Map” doi: null url: https://cs.uwaterloo.ca/journals/JIS/VOL28/Bradshaw/bradshaw3.pdf claim: “Conjecture 20: the solutions to the generalized Collatz commutation problem are squarefree for every pair of parameters of equal parity.” strata_touched:

• D5/S0/Certificates/BradshawConjectureTwentyRefutation license: citation-only triage: anchor

Arithmetic differentiation and the generalized Collatz map

Z. P. Bradshaw, On a Family of Solutions to Arithmetic Differential Equations Involving the Collatz Map, Journal of Integer Sequences 28 (2025), Article 25.1.8.

Printed page 3 defines the arithmetic derivative:

We define the arithmetic derivative to be a non-linear derivation $D:\mathbb{N}\to \mathbb{N}$ on the set of natural numbers with the property that $D(1)=D(0)=0$ and $D(p)=1$ for all primes $p$. Explicitly, we define $D$ so that $D(mn)=D(m)n+mD(n)$ (1) for every $m,n\in \mathbb{N}$.

Printed page 16 defines the generalized Collatz functions:

It should also be noted that similar problems can be formulated for the generalized Collatz functions $C_{a,b}(n):=\{\begin{array}{ll}\frac{an+b}{2},& \text{if }n\text{ odd};\\ \frac{n}{2},& \text{if }n\text{ even},\end{array}$ with $a\equiv b(\mathrm{mod}2)$.

Conjecture 20, printed page 16:

The solutions to the commutation problem $D(C_{a,b}(n))=C_{a,b}(D(n))$ are squarefree for every $a\equiv b(\mathrm{mod}2)$.

The formal predicate lists the four derivative properties in the order $D(0)=0$, $D(1)=0$, prime values, and the product rule. The formal map uses natural-number integer division and the odd test n % 2 = 1. For parameters of equal parity, each selected branch agrees with the source formula.

The encoded claim quantifies over arithmetic derivatives and natural $a,b,n$ of equal parameter parity. The extra hypothesis $1\le n$ makes the claim weaker than the source statement, so refuting it refutes the source, and excludes the trivial $n=0$ boundary. At $(a,b,n)=(17,7,125)$ both sides of the commutation equation equal $641$, but $125=5^3$ is not squarefree. These parameters are positive and odd with $b<a$, as in the five pairs tested in the source.

Verified locator

• URL: https://cs.uwaterloo.ca/journals/JIS/VOL28/Bradshaw/bradshaw3.pdf
• Journal of Integer Sequences 28 (2025), Article 25.1.8.
• Printed page 3: Section 2, definition of the arithmetic derivative, equation (1).
• Printed page 16: generalized Collatz functions and Conjecture 20.
• The article has no DOI; the journal URL is the stable locator.