David Bařina's 7x+1 conjecture

David Bařina's mod 8 conjecture

The iterates x, t(x), t(t(x)),... of the mapping
\[ t(x)= \left\{ \begin{array}{cl} 5x+1 & \mbox{ if $x\equiv -1\pmod{8}$}\\ 5x-1 & \mbox{ if $x\equiv 1\pmod{8}$}\\ \left\lfloor x/2\right\rfloor & \mbox{ otherwise} \end{array} \right. \] 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 Bařina 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 5^3/23(1/2)^4/23(1/2)^2/23(1/2)^5/23(1/2)^3/23(1/2)^4/235^2/23 < 1

⇔ (1/2)^{(4+2+5+3+4)/23}5^5/23 < 1

⇔ 5⁵ < 2¹⁸ ⇔ 3125 < 262144,

thereby predicting everywhere eventual cycling. The cycles found are

0
-1, -4, -2
1, 4, 2
-31, -156, -78, -39, -196, -98, -49, -244, -122, -61.

Last modified 27th July 2018
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