David Barina's mod 8 conjecture

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

t(x) = 5x+1 if x ≡ -1 (mod 8)
t(x) = 5x-1 if x ≡ 1 (mod 8)
t(x) = ⌊x/2⌋ otherwise

are conjectured to eventually reach 1 if x > 0, or one of -1 or -31 if x < 0.

This phenomenon was communicated to Keith Matthews by David Barina on July 13, 2018.

The mapping can be regarded as an 8-branched example of type (b).

Here Q(8) has the zero class as a transient class and the submatrix formed by deleting rows 1 and column 1 has stationary vector (1/23)×(3,4,2,5,3,4,2).

The corresponding weighted product is 53/23(1/2)4/23(1/2)2/23(1/2)5/23(1/2)3/23(1/2)4/2352/23 < 1

⇔ (1/2)(4+2+5+3+4)/2355/23 < 1

⇔ 55 < 218 ⇔ 3125 < 262144,

thereby predicting everywhere eventual cycling. The cycles found are

Enter x (≠ 0):

Last modified 27th July 2018
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