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. ERRATUM AND REVISION NOTICE (v5. 5, 14 June 2026) This version is a corrective revision of v5. 4 following an external review by an automated proof assistant. WHAT WAS WRONG IN v5. 4 The proof of Corollary 11 in v5. 4 contained an incorrect monotonicity argument: CRT does not decompose into coordinate-wise comparisons. A sign flip that raises one residue can decrease the overall CRT value after reduction modulo Mₖ. As a consequence, lower-half tuples do not in general coincide with orbit minima. Numerical check: at k = 6, only 10 of 180 lower-half tuples are orbit minima, so the v5. 4 algorithm would have recovered only 1 of the 33 primes in (13, 169]; at k = 8, only 3 of the 64 primes in (19, 361]. WHAT IS FIXED IN v5. 5 1. Corollary 11 reformulated: lower-half tuples Tₖ parametrise Gₖ-orbits bijectively, but the orbit minimum is now computed by an explicit inner minimum over all 2^ (k-1) sign patterns through CRT. 2. New Remark 12 with a concrete counterexample (k = 6, tuple (1, 1, 1, 1, 2) has CRT representative 6931 but orbit minimum 769). 3. Pseudocode updated with the inner loop over sign patterns. 4. Off-by-one in the counting formulas corrected: number of orbits is phi (Mₖ) /2^ (k-1), so L₆ = 180 (not 90), L₈ = 12960 (not 6480). Total cost per stage is phi (Mₖ) CRT operations, growing as Mₖ / ln pₖ. 5. The Concrete numbers section, the cost discussion, and the final remark on elimination methods have been updated accordingly. WHAT REMAINS UNAFFECTED Theorem 8 (the main identification of primes with orbit minima in (pₖ, pₖ²] for k >= 6) is structurally independent of the error. Its proof uses Lemma 3, Lemma 7, Parts A and B, and Lemma 6, none of which invoke the lower-half characterisation. Theorem 17, Lemma 16, and the appendix verification tables are also unaffected; the tables in Appendices A and B were produced by direct computation rather than from the erroneous formula. The author thanks the external reviewer for the careful catch. 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 (Sun,) studied this question.