Distribution of Prime Numbers in a Grid

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? 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. 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. 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?