This paper presents a closed-form algorithm that produces, for any integer k >= 6, the complete and ordered list of prime numbers in the interval (pₖ, pₖ²], where pₖ denotes the k-th prime. The algorithm operates on the unit group U₌䂵 of the k-th primorial Mₖ = p₁ * p₂ *. . . * pₖ, under the canonical action of the elementary abelian group Gₖ = (Z/2) ^k-1 via sign flips on the odd prime components fixed by the Chinese Remainder Theorem. The orbit of an element y is characterized algebraically by z in U₌䂵: z² = y² mod Mₖ. The main theorem establishes that for k >= 6 the primes in (pₖ, pₖ²] are exactly the orbit minima of this action that fall into the interval. Sorting the orbit minima yields the primes in their natural increasing order. PROOF METHOD The proof relies only on elementary number theory: the Chinese Remainder Theorem, divisor-product bounds, 2-adic valuations, and Bertrand's Postulate. No analytic methods, no zeta function machinery, no character sums. SCOPE AND LIMITATIONS What is fully proven: - Theorem 8: primes in (pₖ, pₖ²] = orbit minima in (pₖ, pₖ²], for all k >= 6. - Theorem 17: smallest composite orbit minimum in (pₖ², Mₖ/2] = p₊+₁², for all k >= 6. - Closed-form enumeration of all primes greater than 13 via the ladder of stages. What is honestly left open: - A full characterization of all composite orbit minima above p₊+₁² (Remark 19). The condition "all prime factors greater than pₖ" is necessary but not sufficient; at k = 6 this condition is satisfied by 2846 integers while only 27 are orbit minima. - Whether every nontrivial orbit of Gₖ on U₌䂵 contains at least one prime (Observation 20). Verified numerically for k = 6, 7 but not proved. RELATION TO CLASSICAL METHODS The construction is not a sieve. Each prime is produced bottom-up from a tuple of residue choices via the Chinese Remainder Theorem, not by eliminating candidates from a list. The interval (pₖ, pₖ²] is selected because it is the precise range in which Lemma 7 (every composite unit modulo Mₖ has prime factors greater than pₖ) forces the desired identification. COMPUTATIONAL COMPLEXITY The classical cost is exponential in k: stage k performs phi (Mₖ) /2ᵏ Chinese-Remainder-Theorem computations, which grows as Mₖ / (2ᵏ * ln pₖ). The closing discussion notes that quantum implementations of modular arithmetic and orbit enumeration would change this cost profile in a structured way.
Thomas Krause (Wed,) studied this question.