On integer linear combinations of terms of rational cycles for the generalized 3x+1 problem

TL;DR

This paper investigates linear combinations of rational cycle terms in generalized Collatz sequences, proving conditions under which these combinations are integers and exploring implications for digit patterns in p-adic representations.

Contribution

It introduces new conditions for linear combinations of cycle terms to be integers and connects these results to digit patterns in p-adic representations.

Findings

Certain linear combinations are integers under specific coefficient conditions

Examples demonstrate the theoretical results

Results help explain digit patterns in p-adic representations

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.