### David Barina's 7x+1 conjecture

The iterates x, t(x), t(t(x)),... of the mapping

t(x) | = | x/2 | if x ≡ 0 (mod 4) |

t(x) | = | 7x+1 | if x ≡ 1 (mod 4) |

t(x) | = | x/2 | if x ≡ 2 (mod 4) |

t(x) | = | 7x-1 | if x ≡ -1 (mod 4) |

are conjectured to eventually reach 1 if x > 0, or -1 if x < 0.
This remarkable phenomenon was communicated to Keith Matthews by David Barina on July 6, 2018. See his paper for an heuristic explanation of this phenomenon.

We remark that the iterates of -x are the negative of the iterates of x, so it is enough to consider positive starting values x.

The mapping is an example of type (b). Here the associated Markov matrix Q(4) has stationary vector (1/2, 1/8, 1/4, 1/8) and we have the inequality

(1/2)^{1/2} 7^{1/8} (1/2)^{1/4} 7^{1/8} < 1,

thereby predicting everywhere eventual cycling.

*Last modified 7th July 2018 *

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