Math 2.0

Number theory Sequences and sets of integers Conjecture

Collatz conjecture

Builds on Natural numbersSequenceNatural density

Statement

Let TT be the map on the positive integers given by T(n)=n/2T(n) = n/2 if nn is even and T(n)=3n+1T(n) = 3n + 1 if nn is odd. For every positive integer nn, there is an integer k≥0k \ge 0 with Tk(n)=1T^k(n) = 1, where TkT^k is the kk-fold iterate of TT.

Remarks

The conjecture is open. It has been checked by computer for all starting values up to very large bounds. Riho Terras showed in 1976 that the set of nn having some iterate Tk(n)<nT^k(n) < n has natural density 11. Terence Tao showed in 2019 that almost all orbits, in the sense of logarithmic density, reach values below f(n)f(n) for any function ff that tends to infinity. For example, starting from 66 the sequence is 6,3,10,5,16,8,4,2,16, 3, 10, 5, 16, 8, 4, 2, 1. Starting from 2727 it takes 111111 steps to reach 11 and climbs as high as 92329232 on the way. Once the sequence reaches 11, it cycles through 1,4,2,11, 4, 2, 1; the conjecture says that this is the only cycle and that no orbit grows without bound.

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Natural density, concept, no evidence levelNatural densityconceptNatural numbers, object, no evidence levelNatural numbersobjectSequence, concept, no evidence levelSequenceconceptCollatz conjecture, conjecture, no evidence levelCollatz conjectureconjecture
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  1. Natural density concept
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  4. Collatz conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Collatz_conjecture
About this page
Tag6PCQEXEX4N
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Kindconjecture
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MSC11B83, 11B37
StatusThis copy of the site has no build data. Run m2 status 6PCQEXEX4N.
AboutCollatz conjecture
UsesNatural numbers, Sequence, Natural density
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Changed2 November 2026

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