Purpose of Day 02
Uniform notation is required before further computational or theoretical work. This page fixes the definitions that will be used throughout the 90-day program. No new theorems or computations are claimed; every statement is either a definition (Kanıtlandı) or an attribution of standard literature (Bilinen literatür).
1. The Original Collatz Map
Definition 1.1 Kanıtlandı
The original Collatz map \(C:\mathbb{N}\to\mathbb{N}\) is defined by
This is the classical formulation. Every odd step is followed by at least one even step because \(3n+1\) is even for odd \(n\).
2. The Accelerated Terras Map \(T\)
Definition 1.2 Kanıtlandı
The accelerated (Terras) map \(T:\mathbb{N}\to\mathbb{N}\) folds the forced division by 2 into the odd branch:
This is the map used for all verification and stopping-time computations on this site. It is equivalent to the original map for the purpose of reaching 1, but counts fewer steps.
Bilinen literatür Terras (1976) introduced the stopping-time analysis for this accelerated formulation.
3. The Syracuse Map
Definition 1.3 Kanıtlandı
Let \(\mathcal{O}=2\mathbb{N}+1\) be the odd positive integers. The Syracuse map \(S:\mathcal{O}\to\mathcal{O}\) is defined by
where \(v_2\) is the 2-adic valuation (the largest exponent of 2 dividing the argument). Equivalently, \(S(n)\) is the largest odd divisor of \(3n+1\).
Iterating \(S\) produces the sequence of successive odd terms in a Collatz trajectory. The Collatz conjecture is equivalent to the statement that every odd starting value eventually reaches 1 under \(S\).
Bilinen literatür The Syracuse formulation is standard (see Lagarias surveys and Tao 2019).
4. Stopping Time and Total Stopping Time
Definition 1.4 Kanıtlandı
For \(n\ge 2\), the stopping time \(\sigma(n)\) (also denoted \(\chi(n)\) by Terras) is the least positive integer \(k\) such that \(T^k(n) The Collatz conjecture is equivalent to \(\sigma(n)<\infty\) for all \(n\ge 2\). The total stopping time \(\sigma_\infty(n)\) is the least positive integer \(k\) such that \(T^k(n)=1\) (or equivalently that the orbit reaches the cycle \(\{1,2\}\)). Note that \(\sigma(n)\le\sigma_\infty(n)\) whenever both are finite. The verification performed on this site establishes that \(\sigma(n)<\infty\) for all \(2\le n<2^{36}\). Bilinen literatür Terras (1976) proved that the set of \(n\) with finite stopping time has asymptotic density 1. For the original (unaccelerated) map \(C\), the glide of \(n\) is the analog of the stopping time: the least number of iterations of \(C\) until the trajectory falls strictly below its starting value. In the literature this is sometimes called the height or the length of the ascent phase before the first descent. On this site, primary statistics are reported for the accelerated map \(T\). Glide lengths for the original map are noted only when comparing to external records. The coefficient stopping time \(\tau(n)\) is the least \(k\) such that the product of the multiplicative factors (ignoring the additive \(+1\) terms) falls below 1, i.e., \(3^{q}/2^{k}<1\) where \(q\) is the number of odd steps in the first \(k\) iterations. Terras conjectured that \(\sigma(n)=\tau(n)\) for all \(n\ge 2\). This remains open and is independent of the main Collatz conjecture, though it would imply the absence of nontrivial cycles.Definition 1.5 Kanıtlandı
5. Glide
Definition 1.6 Bilinen literatür
6. Coefficient Stopping Time (Terras)
Definition 1.7 Bilinen literatür
7. Summary Table of Notation
Symbol Meaning Label \(C(n)\) Original Collatz map Kanıtlandı \(T(n)\) Accelerated Terras map Kanıtlandı \(S(n)\) Syracuse map (odd-to-odd) Kanıtlandı \(\sigma(n)\) Stopping time under \(T\) Kanıtlandı \(\sigma_\infty(n)\) Total stopping time to 1 Kanıtlandı glide First-descent length under original \(C\) Bilinen literatür \(\tau(n)\) Coefficient stopping time Bilinen literatür 8. Relation to the Conjecture