$3n + 3^k$ Problem

David Barina; W. C. Maat

$3n + 3^k$ Problem

David Barina, W. C. Maat

Abstract

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 $3^k$.

$3n + 3^k$ Problem

Abstract

The Collatz problem is generalized into

problem. It is shown that as long as the Collatz function iterates converge to the cycle passing through the number 1, the

sequence converges to the cycle passing through the number

for arbitrary positive integers

and

. The proof shows that the sequence of

function iterates for a number

is exactly the sequence of the Collatz function iterates for

multiplied by

Paper Structure (2 sections, 6 theorems, 22 equations, 2 tables)

This paper contains 2 sections, 6 theorems, 22 equations, 2 tables.

Key Result

Theorem 1

As long as the $T_0$ iterates converge to 1, a sequence defined by repeatedly applying the function converges to the cycle passing through the number $3^k$ for arbitrary positive integers $n$ and $k$.

Theorems & Definitions (10)

Theorem 1
proof
Corollary 1.1
Corollary 1.2
Lemma 1.1
proof
Corollary 1.3
proof
Theorem 2
proof