2026/07/24 by Joseph M. Shunia
Mathematics · #math.NT
Starting from a positive integer n and no a priori information about the primes above it, we construct a polynomial quotient ring that recovers exactly the primes in (n,2n] from a single modular exponentiation. The primes occur simultaneously as the nonzero monomial degrees of the resulting polynomial remainder, and each coefficient independently certifies its corresponding prime through its additive order. When n=pk is prime, the least nonzero degree is pk+1. Thus the next prime is recovered from the preceding prime alone, without using the index k, the prime-counting function, a prime table, nor any primality tests. We develop the underlying ring structure, give equivalent annihilator and quotient formulations, extend the result to shorter intervals, and provide a SageMath implementation.