DOI: 10.5281/zenodo.14837983
title: Data set showing positions where prime numbers do not occur
title: Data set showing positions where prime numbers do not occur.

description: This study presents the discovery of a recurring pattern in non-prime odd numbers, verified both visually and computationally up to 1,000,000. The research utilized Python to structure and validate the organization of numbers in blocks, highlighting positions where prime numbers never occur. The attached materials include the explanatory article, the spreadsheet with the distribution of patterns, and the source code used for the analysis.
description: Discovering Patterns in Non-Prime Odd Numbers

Introduction

Mathematics has always fascinated humanity with its ability to reveal hidden patterns in numbers. This study, conducted by Robson Lemos Vargas, a person diagnosed with Autism Spectrum Disorder (ASD), presents a visual and computational analysis that led to the discovery of a recurring pattern in non-prime odd numbers. The pattern was verified for numbers between 1 and 1,000,000 using Python, generating an explanatory table in a spreadsheet.

Structure and Organization of the Table

The arrangement of numbers in the table follows a specific organization based on blocks. Each block is structured into columns and rows that follow an ordered progression. The numbering progresses in tens, distributed according to an alternating sequence of 1, 3, 7, and 9, repeated successively in each column.

Identified Visual Pattern

Within each block, the structure reveals a zigzag visual among numbers that follow a recurring pattern. The observed pattern starts with the numbers 21 and 27, then 33 and 39, repeating horizontally along the columns. After each set, a column is skipped, and the sequence continues in the corresponding positions. This regular behavior is observed up to 1 million.

Besides the horizontal pattern, there is also a vertical pattern identified in the columns, where certain positions always remain without the presence of prime numbers. All other positions that do not follow this horizontal pattern were marked in the spreadsheet with blue-colored cells. To facilitate visualization, the attached image presents this pattern with differentiated colors, highlighting the structural alignments formed in the number arrangement.

Methodology

Building the Code and Table

The pattern was visually identified and later structured in a Python code for validation and extended analysis. The code:





Identifies prime numbers, applying red formatting and bold font.




Highlights non-prime odd numbers within the pattern, marking their cells in blue in the spreadsheet and in green in the attached image.




Excludes even numbers, allowing exclusive focus on odd numbers.




Organizes numbers into vertical blocks, ensuring that the distribution follows a logical and consistent progression.



The code was executed to generate a spreadsheet that verified the pattern up to 1,000,000. The generated file is self-explanatory, with colors and formatting highlighting the described patterns.

Results

The pattern of non-prime odd numbers was confirmed to be recurring in number blocks throughout the analyzed range (1 to 1,000,000). This pattern completely excludes prime numbers, which never occupy the highlighted positions, validating the consistency of the initial observation.

Additionally, numbers in the identified columns as part of the pattern (such as 341, 671, 1001, etc.) also follow this recurrence, suggesting that there is an underlying modular structure governing this arrangement.

The analysis confirms that there are no other recurring positions where primes never appear. After extensive testing and observation, all possible variations were considered, and the structure remained stable and predictable within the studied range.

Discussion

Although this study does not intend to provide a rigorous mathematical explanation for the identified pattern, it offers a starting point for future investigations. The organization of blocks, following ordered progressions and evidencing predictable graphical patterns, raises questions about the relationship between composite numbers and their spatial distribution within odd numbers.

The number 2 was excluded from the study for being the only even prime, and the numbers 1, 3, 5, and 9 were also omitted from the table due to an initial misalignment that prevents the regularity of the pattern in the first elements of the sequence. This initial misalignment is similar to the situation of the number 11, which, in the first block of the table, does not follow the pattern of the other marked cells, but in subsequent blocks aligns with the identified pattern.

Additionally, the explanation based on modular structure reinforces that the highlighted numbers will never be prime due to their arithmetic nature, creating a fertile field for exploring similar patterns in different progressions.

Conclusion

The spreadsheet and Python code developed for this study are available for analysis by mathematicians and other interested parties. It is hoped that the simplicity of the approach and the clarity of the results will encourage collaboration and new discoveries. To access the files:





Python source code: Attached to the study.




Spreadsheet with the pattern distribution: Attached to the study.




Image with the graphical representation of the pattern: Attached to the study.



Robson Lemos Vargas invites mathematicians and interested scholars to explore this pattern in greater depth, thus contributing to the advancement of mathematical understanding.