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MONOTONIC COLLATZ SUBSEQUENCES WITH TERMS CONGRUENT MODULO A FIXED POWER OF TWO
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 16 May 2025, pp. 1-12
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- Article
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A class of sequences called L-sequences is introduced, each one being a subsequence of a Collatz sequence. Every ordered pair
$(v,w)$ of positive integers determines an odd positive integer P such that there exists an L-sequence of length n for every positive integer n, each term of which is congruent to P modulo 
$2^{v+w+1}$. The smallest possible initial term of such a sequence is described. If 
$3^v>2^{v+w}$ the L-sequence is increasing. Otherwise, it is decreasing, except if it is the constant sequence P. A central role is played by Bezout’s identity.
