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# Almost all Collatz orbits attain almost bounded values What's new

I’ve just uploaded to the arXiv my paper “Almost all Collatz orbits attain almost bounded values“, submitted to the proceedings of the Forum of Mathematics, Pi. In this paper I returned to the topic of the notorious Collatz conjecture (also known as the \$latex {3x+1}&fg=000000\$ conjecture), which I previously discussed in this blog post. This conjecture can be phrased as follows. Let \$latex {{bf N}+1 = {1,2,dots}}&fg=000000\$ denote the positive integers (with \$latex {{bf N} ={0,1,2,dots}}&fg=000000\$ the natural numbers), and let \$latex {mathrm{Col}: {bf N}+1 rightarrow {bf N}+1}&fg=000000\$ be the map defined by setting \$latex {mathrm{Col}(N)}&fg=000000\$ equal to \$latex {3N+1}&fg=000000\$ when \$latex {N}&fg=000000\$ is odd and \$latex {N/2}&fg=000000\$ when \$latex {N}&fg=000000\$ is even. Let \$latex {mathrm{Col}_{min}(N) := inf_{n in {bf N}} mathrm{Col}^n(N)}&fg=000000\$ be the minimal element of the Collatz orbit \$latex {N, mathrm{Col}(N), mathrm{Col}^2(N),dots}&fg=000000\$. Then we have

Conjecture 1 (Collatz conjecture) One has \$latex {mathrm{Col}_{min}(N)=1}&fg=000000\$ for…

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