A Pythagorean Sieve and a Recursive Modular Characterization of Odd Primes

Starting from the elementary geometric identity r =(x − d)/2, valid for integer sides x and divisors d of x 2 with 1 < d < x, and yielding an integer inradius r whenever x is odd, we construct a new sieve procedure acting on the variable r. This sieve reproduces, within a Pythagorean framework, the mechanism of Eratosthenes' sieve and yields a complete characterization and enumeration of the odd prime numbers. The resulting modular system admits a recursive structure (Mt, St) analogous to a wheel factorization, thereby establishing a direct correspondence between the arithmetic of primes and the geometry of integer right triangles. Although sieve-based in its foundation, the method also functions as an ordered generator of the odd primes via the relation pk = 2rk + 1. A concise Python implementation of the recursive sieve on r, included in Appendix A,illustrates its computational realization and veries the theoretical results. In particular, the ordered nature of the sieve enables the direct computation of the n-th prime through the routine nth_prime_r(n), without precomputing all preceding primes.