[Paper Review] Analysis of Prime Reciprocal Sequences in Base 10

Sumanth Kumar Reddy Gangasani|ArXiv.org|Jan 24, 2008

Coding theory and cryptography11 references3 citations

TL;DR

This paper analyzes the digit-level structure of prime reciprocal sequences in base 10, classifying recurring patterns based on the least significant digit of the prime. It reveals that digit 0 is the most frequent across all such sequences, highlighting structural redundancy relevant to cryptographic and pseudorandom sequence generation applications.

ABSTRACT

Prime reciprocals have applications in coding and cryptography and for generation of random sequences. This paper investigates the structural redundancy of prime reciprocals in base 10 in a manner that parallels an earlier study for binary prime reciprocals. Several different kinds of structural relationships amongst the digits in reciprocal sequences are classified with respect to the digit in the least significant place of the prime. It is also shown that the frequency of digit 0 exceeds that of every other digit when the entire set of prime reciprocal sequences is considered.

Motivation & Objective

• To investigate structural redundancies in prime reciprocal sequences when expressed in base 10.
• To classify recurring digit patterns based on the least significant digit of the prime divisor.
• To quantify digit frequency distributions across the full set of prime reciprocal sequences.
• To assess implications for applications in coding theory and cryptography, particularly in pseudorandom sequence generation.

Proposed method

• The study analyzes the decimal expansions of 1/p for all prime numbers p > 10.
• It categorizes recurring digit patterns according to the units digit (1, 3, 7, 9) of the prime p.
• Statistical analysis is performed across all such reciprocals to compute digit frequency distributions.
• The analysis draws parallels to prior work on binary prime reciprocals, extending structural insights to base 10.
• The paper uses empirical enumeration and frequency counting to identify dominant digit behaviors.
• It employs a classification framework to group sequences by their terminal digit, enabling pattern recognition.

Experimental results

Research questions

• RQ1 What structural patterns emerge in the decimal expansions of 1/p for prime p in base 10?
• RQ2 How does the least significant digit of a prime influence the digit distribution in its reciprocal?
• RQ3 What is the relative frequency of each digit (0–9) across all prime reciprocal sequences in base 10?
• RQ4 To what extent is digit 0 more frequent than other digits in these sequences?
• RQ5 How do these structural redundancies affect the suitability of prime reciprocals for cryptographic or pseudorandom sequence generation?

Key findings

• Digit 0 is the most frequently occurring digit in the combined set of all prime reciprocal sequences in base 10.
• The frequency of digit 0 exceeds that of every other digit, indicating a significant structural bias.
• Distinct digit patterns emerge depending on the least significant digit of the prime divisor.
• The study confirms a non-uniform distribution of digits across reciprocal sequences, with 0 dominating.
• The results suggest inherent redundancy in prime reciprocal sequences that may impact their use in random sequence generation.
• The findings extend prior binary-based analyses to base 10, revealing consistent structural tendencies across numeral systems.