### Conjecture of Pruthviraj Hajari

This generalizes Hahari's first conjecture.
Let k ≥ 3 be an odd integer. The iterates x, t(x), t(t(x)),... of the mapping

t(x) | = | (k^{2} - 1)x | if x is odd |

t(x) | = | ⌊x/k⌋ | if x is even, |

where ⌊ ⌋ denotes the *floor* function.
are conjectured by Pruthviraj Hajari to eventually reach 0 if x ≥ 0. (See MathOverflow question.)
It seems certain that if x < 0, then t^{n}(x) will eventually reach one of the (k^{2} - 1)/4 cycles
with starting points -(ki +2s), where 1 ≤ i ≤ k - 2, t odd, and 0 ≤ s ≤ (k - 1)/2; each cycle is of length 3:

-(ki +2s) → -(ki +2s)(k^{2} - 1) → -k^{2}i - 2sk + i → -(ki +2s).

*Last modified 16th December 2020 *

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