On integer linear combinations of terms of rational cycles for the generalized 3x+1 problem
Yagub N. Aliyev
TL;DR
The paper investigates generalized Collatz sequences defined by linear fractional steps $B_i(x)=(p_i x+k_i)/q$ and studies rational cycles via the $n$-step fixed points $x_i$ and auxiliary quantities $U_i=q^{i}/(q^{n}-p_0p_1\cdots p_{n-1})$. The core result shows that if an integer combination $\alpha U_0+\beta U_b$ holds, then the corresponding cycle terms satisfy an integer linear relation $\alpha x_i+\beta p_i p_{i+1}\cdots p_{i+b-1} x_{i+b}$ for all valid $i$, providing a structural link between $U$-based integrality and the actual cycle terms. This is demonstrated through explicit constructions and proved via intermediate lemmas, with worked examples (including a $q=3$ case) where such combinations evaluate to integers, thereby validating the theory. The authors also connect these integrality properties to $p$-adic representations, showing that suitable choices yield equal $p$-adic digits across shifted terms, and discuss implications for the Finite Cycles Conjecture and possible guidance toward the original $3x+1$ problem. Overall, the work offers a method to capture the arithmetic structure of rational cycles and suggests that integrality of linear combinations can illuminate the digit patterns and finite-cycle phenomena in generalized Collatz dynamics.
Abstract
In the paper, some special linear combinations of the terms of rational cycles of generalized Collatz sequences are studied. It is proved that if the coefficients of the linear combinations satisfy some conditions then these linear combinations are integers. The discussed results are demonstrated on some examples. In some particular cases the obtained results can be used to explain some patterns of digits in $p$-adic representations of the terms of the rational cycles.