Purpose of Day 06
Map the early rigorous literature that introduced the accelerated Collatz map \(T\), defined the stopping time \(\sigma(n)\), and proved that the set of \(n\) with finite stopping time has natural density 1. Every claim is labelled. No claim is made that the conjecture is solved.
1. The Accelerated Map \(T\)
Bilinen literatür Independently observed by Riho Terras (1976) and C. J. Everett (1977) that this form is analytically more convenient than the classical Collatz map \(C\) (which applies the full \(3n+1\) then divides by 2 only later). Iteration of \(T\) simply omits intermediate even steps after odd inputs. See Lagarias overview (arXiv:2111.02635, §1).
Kanıtlandı The two maps generate the same trajectories up to compression of consecutive divisions by 2; the conjecture for one is equivalent to the conjecture for the other.
2. Terras (1976): Stopping Time and Density
Reference: R. Terras, “A stopping time problem on the positive integers,” Acta Arithmetica 30 (1976), 241–252.
Bilinen literatür Terras defined the stopping time
\[
\sigma(n)=\min\{k\ge 1:T^k(n) Bilinen literatür Terras proved that the set of positive integers with finite stopping time has natural density 1. More precisely, the limiting density of integers \(n\le X\) with \(\sigma(n)\le k\) exists for each fixed \(k\) and tends to 1 as \(k\to\infty\). The proof proceeds via the distribution of parity vectors and the central-limit behaviour of the associated multiplicative factors \(3^{q}/2^{k}\). Bilinen literatür He further introduced the coefficient stopping time and conjectured equality with \(\sigma(n)\) (still open in full generality). The density result was obtained independently and contemporaneously by Everett. Almost every positive integer (natural density 1) has finite stopping time under \(T\). This is a rigorous theorem, not a heuristic.Terras density theorem · Bilinen literatür
3. Everett (1977): Almost-All Descent
Reference: C. J. Everett, “Iteration of the number-theoretic function \(f(2n)=n\), \(f(2n+1)=3n+2\),” Advances in Mathematics 25 (1977), 42–45.
Bilinen literatür Everett proved that almost every positive integer \(m\) has some iterate \(f^k(m) Bilinen literatür The paper is short and elegant; it is one of the first research publications specifically devoted to the \(3x+1\) function. Lagarias notes that the result was obtained independently of Terras.
4. Lagarias: Surveys, Bibliography, and the 2010 Volume
Bilinen literatür Jeffrey C. Lagarias has produced the definitive annotated bibliographies:
- “The 3x+1 problem: An annotated bibliography (1963–1999)” (arXiv:math/0309224 and updates).
- “The 3x+1 Problem: An Annotated Bibliography, II (2000–2009)” and later continuations.
Bilinen literatür He edited the volume The Ultimate Challenge: The 3x+1 Problem (AMS, 2010), which reprints the early papers of Terras and Everett with commentary, contains his own overview, and collects the state of knowledge as of 2010. The overview (arXiv:2111.02635) remains the standard modern reference for history and status.
Bilinen literatür Lagarias’s 1985 American Mathematical Monthly survey (“The 3x+1 problem and its generalizations”) already contained the basic modular results and the density statements of Terras–Everett, and formulated the principal open conjectures that are still unsolved.
Kanıtlandı The bibliographic apparatus and the reprinted primary sources establish a stable, citable foundation for any subsequent computational or analytic work.
5. Summary Table of Foundational Claims
| Result | Author(s) | Year | Status | Label |
|---|---|---|---|---|
| Accelerated map \(T\) preferred for analysis | Terras, Everett | 1976–77 | Standard | Bilinen literatür |
| Stopping time \(\sigma(n)\) defined; conjecture \(\Leftrightarrow\sigma(n)<\infty\) | Terras | 1976 | Definition | Bilinen literatür |
| Natural density of finite stopping time = 1 | Terras, Everett | 1976–77 | Theorem | Kanıtlandı |
| Coefficient stopping time equality (CST) | Terras | 1976 | Conjecture | Varsayım/Sezgi |
| Annotated bibliography & overview volume | Lagarias | 1985–2010+ | Reference | Bilinen literatür |
6. Implications for the Present Program
Bilinen literatür The density theorems of Terras and Everett justify the inductive verification strategy used in Days 03–05: once every integer below a bound \(B\) has been shown to descend, the only possible counter-examples lie in a density-zero set whose first members (if any) exceed \(B\). They do not eliminate the possibility of a sparse divergent orbit or a very long cycle.
Varsayım/Sezgi The heuristic geometric-mean factor \(\sqrt{3}/2\approx 0.866<1\) is consistent with the density-1 result but remains non-rigorous for the exceptional set.