Distribution of Prime Numbers in a Grid
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| Natura: | Recurso digital |
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Zenodo
2026
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| _version_ | 1866901516021923840 |
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| author | Allouche Bernard |
| author_facet | Allouche Bernard |
| contents | <p>Ulam's spiral has often been mentioned for its curious ability to reveal diagonal alignments of prime numbers in a square grid. Although this observation is visually striking and continues to fascinate both researchers and countless enthusiasts, the underlying reason remains unclear: why do primes appear more frequently along certain directions?</p> <p><br>Most interpretations rely on complex analytical tools or on sieving techniques. This paper oers a more intuitive perspective, rooted in elementary arithmetic, to explain these diagonal patterns. Our approach builds upon simple congruence relationships between integers placed within a twodimensional structure.</p> <p><br>We present a graphical and systematic method for locating prime numbers inside a regular grid, based on congruence characteris tics modulo dierent integers. Despite its sim plicity, this method leads to structured, sometimes unexpected, geometric properties that echo wellknown number theory concepts.</p> <p><br>Finally, several appendices propose concrete implementations, opening the way for educational exercises as well as amateur explorations. Yet, one very fundamental question remains open: what deeper structure of the integers governs these recurring, almost geometric, appearances of primes?</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18865649 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Distribution of Prime Numbers in a Grid Allouche Bernard Prime number Ulam spiral <p>Ulam's spiral has often been mentioned for its curious ability to reveal diagonal alignments of prime numbers in a square grid. Although this observation is visually striking and continues to fascinate both researchers and countless enthusiasts, the underlying reason remains unclear: why do primes appear more frequently along certain directions?</p> <p><br>Most interpretations rely on complex analytical tools or on sieving techniques. This paper oers a more intuitive perspective, rooted in elementary arithmetic, to explain these diagonal patterns. Our approach builds upon simple congruence relationships between integers placed within a twodimensional structure.</p> <p><br>We present a graphical and systematic method for locating prime numbers inside a regular grid, based on congruence characteris tics modulo dierent integers. Despite its sim plicity, this method leads to structured, sometimes unexpected, geometric properties that echo wellknown number theory concepts.</p> <p><br>Finally, several appendices propose concrete implementations, opening the way for educational exercises as well as amateur explorations. Yet, one very fundamental question remains open: what deeper structure of the integers governs these recurring, almost geometric, appearances of primes?</p> |
| title | Distribution of Prime Numbers in a Grid |
| topic | Prime number Ulam spiral |
| url | https://doi.org/10.5281/zenodo.18865649 |