Papers › Determining monotonic step-equal sequences of any limited length in the Collatz problem
Determining monotonic step-equal sequences of any limited length in the Collatz problem
Longjiang Li
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This paper proposes a formula expression for the well-known Collatz conjecture (or 3x+1 problem), which can pinpoint all the growth points in the orbits of the Collatz map for any natural numbers. The Collatz map Col: 𝒩+1 →𝒩+1 on the positive integers is defined as xₙ₊₁=Col(xₙ)=(3 xₙ +1)/2^(mₙ) where xₙ₊₁ is always odd and mₙ is the step size required to eliminate any possible even values. The Collatz orbit for any positive integer, x₁, is expressed by a sequence, <x₁; x₂≐Col(x₁); ⋯ xₙ₊₁≐Col(xₙ); ⋯> and xₙ is defined as a growth point if Col(xₙ)>xₙ holds, and we show that every growth point is in a format of ``4y+3'' where y is any natural number. Moreover, we derive that, for any given positive integer n, there always exists a natural number, x₁, that starts a monotonic increasing or decreasing Collatz sequence of length n with the same step size. For any given positive integer n, a class of orbits that share the same orbit rhythm of length n can also be determined.
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