Conjecture of Pruthviraj Hajari

This generalizes Hahari's second conjecture.

Let k ≥ 2 be an even integer. The iterates x, t(x), t(t(x)),... of the mapping

t(x) = (k + 1)x + 1 if x is odd
t(x) = ⌈x/k⌉ if x is even,
where ⌈  ⌉  denotes the ceiling function.

Equivalently

t(x) = (k + 1)x+ 1 if x is odd
t(x) = x/k if x ≡ 0 (mod k)
t(x) = ⌊x/k⌋ + 1 if x ≡ 0 (mod 2) and x ≢ 0 (mod k),

are conjectured by Pruthviraj Hajari to eventually reach 1 if x > 0. (See MathOverflow question.)

It seems certain that if x ≠ 0, then tn(x) will eventually reach one of the cycles 0 → 0, or 1 → k + 2 → 2 → 1, or -(2t + 1) → -2t(k + 1) - k → -(2t + 1), where 0 ≤ t ≤ k/2 - 1.

Enter k (2 ≤ k ≤ 100 and even) :
Enter x :

Last modified 12th December 2020
Return to main page