On the Uniformity of $(3/2)^n$ Modulo 1

TL;DR

This paper investigates the distribution of the sequence (3/2)^n modulo 1, providing computational evidence supporting the conjecture of its uniform distribution, which has implications for number theory problems like the Collatz conjecture.

Contribution

It introduces an efficient algorithm to compute (3/2)^n modulo 1 for very large n and offers statistical analysis supporting the uniformity hypothesis.

Findings

Sequence (3/2)^n modulo 1 appears uniformly distributed

Algorithm efficiently computes (3/2)^n modulo 1 for n up to 10^8

Results support the conjecture of uniform distribution in this sequence

Abstract

It has been conjectured that the sequence $(3/2)^n$ modulo $1$ is uniformly distributed. The distribution of this sequence is signifcant in relation to unsolved problems in number theory including the Collatz conjecture. In this paper, we describe an algorithm to compute $(3/2)^n$ modulo $1$ to $n = 10^8$. We then statistically analyze its distribution. Our results strongly agree with the hypothesis that $(3/2)^n$ modulo 1 is uniformly distributed.