The length of the repeating decimal

TL;DR

This paper explores how to determine the length of the repeating decimal part of fractions, especially when denominators are prime powers, and discusses related number theory problems including cyclic numbers and Gauss's question.

Contribution

It provides a method to compute the repeating decimal length via prime factorization and analyzes conditions for maximum length, connecting to longstanding mathematical questions.

Findings

Repeating decimal length is the least common multiple of prime factor lengths.

Conditions for repeating decimal length to be denominator minus one are characterized.

The existence of infinitely many such fractions is discussed.

Abstract

This paper investigates the length of the repeating decimal part when a fraction is expressed in decimal form. First, it provides a detailed explanation of how to calculate the length of the repeating decimal when the denominator of the fraction is a power of a prime number. Then, by factorizing the denominator into its prime factors and determining the repeating decimal length for each prime factor, the paper concludes that the overall repeating decimal length is the least common multiple of these lengths. Furthermore, it examines the conditions under which the repeating decimal length equals the denominator minus 1 and discusses whether such fractions exist in infinite quantity. This topic is connected to an unsolved problem posed by Gauss in the 18th century and is also closely related to the important question of whether cyclic numbers exist in infinite quantity.