$3n + 3^k$ Problem

The Collatz problem is generalized into $3n + 3^k$ problem. It is shown that as long as the Collatz function iterates converge to the cycle passing through the number 1, the $3n + 3^k$ sequence converges to the cycle passing through the number $3^k$ for arbitrary positive integers $n$ and $k$. The proof shows that the sequence of $3n + 3^k$ function iterates for a number $3^k n$ is exactly the sequence of the Collatz function iterates for $n$ multiplied by $3^k$.

💡 Research Summary

The paper “3n + 3ᵏ Problem” proposes a straightforward generalisation of the classic Collatz map. The authors define a family of maps Tₖ : ℕ→ℕ by

  Tₖ(n) = (3n + 3ᵏ)/2 if n is odd,
  Tₖ(n) = n/2      if n is even,

and they denote the original Collatz map by T₀ (the case k = 0). The central claim (Theorem 1) is that, assuming the Collatz conjecture holds for T₀ (i.e., every orbit reaches the 1‑2‑1 cycle), every orbit of Tₖ reaches the analogous cycle containing 3ᵏ, namely 3ᵏ → 2·3ᵏ → 3ᵏ.

The proof rests on a scaling operator L₍₃ᵏ₎(n)=n/3ᵏ (with inverse L₍₃ᵏ₎⁻¹(n)=3ᵏ·n). The authors observe that

  Tₖ = L₍₃ᵏ₎⁻¹ ∘ T₀ ∘ L₍₃ᵏ₎,

which immediately yields the exact relationship

  Tₖ(3ᵏ·n) = 3ᵏ·T₀(n)

for all n. Consequently, the whole orbit of 3ᵏ·n under Tₖ is simply the orbit of n under T₀ multiplied by 3ᵏ. This scaling argument is illustrated with explicit tables for k = 0, 1, 2, showing how the sequences line up after multiplication by 3ᵏ.

Corollary 1.1 formalises the cycle structure: Tₖ’s integer cycles are precisely the 1‑2‑1 cycle of T₀ stretched by a factor 3ᵏ, i.e. 1·3ᵏ → 2·3ᵏ → 1·3ᵏ. Corollary 1.2 restates the scaling identity, while Corollary 1.3 proves that any integer cycle of Tₖ corresponds to an integer cycle of T₀ multiplied by 3ᵏ, and vice‑versa. The proof of Corollary 1.3 relies on a result of Lagarias (1990) that the Collatz map has no rational cycles whose denominator is a multiple of 3. By showing that any rational cycle of Tₖ would induce a rational cycle of T₀ with denominator divisible by 3, the authors conclude that such cycles cannot exist, forcing any integer cycle of Tₖ to be a 3ᵏ multiple of a T₀ cycle.

The paper then broadens the scope to maps of the form

  T_O(n) = (3n + O)/2 if n odd,
  T_O(n) = n/2     if n even,

where O ≡ ±1 (mod 6). Using the same scaling operator, they prove (Theorem 2) that the integer cycles of T_{3ᵏ O} are exactly the integer cycles of T_O multiplied by 3ᵏ. The algebraic manipulations exploit the commutativity of scaling operators (L_A ∘ L_B = L_{AB}) and again invoke Lagarias’s rational‑cycle theorem to rule out unwanted denominators.

In the conclusion the authors argue that studying Tₖ does not yield new information about the original Collatz problem because Tₖ’s dynamics on multiples of 3ᵏ are identical to T₀’s dynamics, and any integer eventually becomes a multiple of 3ᵏ after finitely many steps. Nevertheless, they note that a proof of convergence for any Tₖ would automatically settle the original conjecture, and that all heuristic arguments for T_O carry over to T_{3ᵏ O}.

Overall, the paper provides a clean algebraic observation: the 3n + 3ᵏ map is a simple rescaling of the classic Collatz map. The argument is mathematically correct under the assumption that the Collatz conjecture holds, but it does not advance the conjecture itself. The contribution is primarily expository, clarifying how a whole family of “scaled” Collatz maps inherit the same orbit structure. The presentation suffers from typographical errors and occasional gaps in rigor (e.g., the handling of infinite powers of two before reaching a multiple of 3ᵏ), but the central scaling insight is sound and may be useful for researchers exploring variants of the Collatz dynamics.