The Collatz conjecture asks whether the following procedure always terminates at 1. Take any positive integer *n*. If it’s odd, multiply it by 3 and add 1. Otherwise, divide it by 2. For obvious reasons the Collatz conjecture is also known as the 3*n* + 1 conjecture. It has been computationally verified that the Collatz conjecture is true for *n* less than 2^{68}.

Alex Kontorovich posted a long thread on Twitter last week explaining why he thinks the Collatz conjecture might be false. His post came a few days after Tao’s paper was posted giving new partial results toward proving the Collatz conjecture is true.

Tao cites earlier work by Kontorovich. The two authors don’t contradict each other. Both are interested in the Collatz problem and have produced very technical papers studying the problem from different angles. See Konotorovich’s papers from 2006 and 2009.

Konotorovich’s 2009 paper looks at both the 3*n* + 1 problem and the 5*n* + 1 problem, the latter simply replacing the “3” in the Collatz conjecture with a “5”. The 5*n* + 1 has cycles, such as {13, 33, 83, 208, 104, 52, 26}. It is conjectured that the sequence produced by the 5*n* + 1 problem diverges for almost all inputs.

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[1] Thanks for the update in the comments.

The 2^60 limit is a very old result. T. Oliveira e Silva overtook it already in 2009 [1]. Later, in 2017, the yoyo@home project verified all numbers below ca. 2^66 [2]. Recently, D. Barina raised this limit to 2^68 [3].

References:

[1] T. Oliveira e Silva. Maximum excursion and stopping time record-holders for the 3x + 1 problem: Computational results. Mathematics of Computation, 68(225):371–384, 1999.

[2] C. Hercher. Über die Länge nicht-trivialer Collatz-Zyklen. Die Wurzel, 6 and 7, 2018.

[3] D. Barina. Convergence verification of the Collatz problem. The Journal of Supercomputing, 2020.