A fractal algorithm shows prime number patterning

We present a novel fractal algorithm that models prime number distribution via recursively generated symbolic sieves structured by primorial-based periodic patterns. This approach constructs self-similar “envelopes”—families of arithmetic progressions containing all prime candidates and semiprimes—with prime gaps and composite patterns emerging through fractal, recursive transformations. Analysis reveals the prime distribution’s fractal dimension approaches but remains below one, reflecting intricate, near line-filling structure. The algorithm offers new deterministic insights into prime gaps, factorization, and efficient large prime generation, complementing classical analytic number theory by uncovering the hidden fractal architecture governing prime patterns.