Relational Inversions of Deterministic Primality Certificate Theorems as a Road Map for Algebraic Decomposition of the Primes: A Demonstration via Inversion of Pocklington's Theorem

Abstract 

Traditional computational number theory relies on top-down, blank-slate validation filters to determine the primality of an isolated integer *N*, incurring a computational penalty that scales with the magnitude of the candidate space. This paper introduces a novel method: the relational inversion of a deterministic primality certificate theorem, by which the asymmetries latent in a verification theorem are identified and exploited to construct a prime generating algorithm. The prime hard limit of that algorithm reveals a discrete set of numbers that are simultaneously genuine primes and members of a previously unidentified algebraic subset within ℙ. Pocklington's Theorem is inverted here as the demonstration instance of this method.

Rather than testing arbitrary candidates, a known prime *p* is fixed as base prime from which candidate pairs *N* = *p* · *U* ± 1 are constructed with even multiplier *U* < *p*. This arithmetic constraint guarantees Pocklington's factorization requirement *F* > √*N* a priori, eliminating the traditional factorization search entirely. By evaluating a reduced primality criterion derived directly from Pocklington's two conditions, verification complexity is reduced from 𝒪(√*N*) to 𝒪(log *U* + log *p*) per candidate. As the algorithm generates successive certified primes, this complexity advantage increases exponentially.

The algorithm is fully deterministic: every integer it certifies is a guaranteed prime by Pocklington's Theorem, with zero false positives across all tested candidates. Integers outside the algorithm's certification set are returned as *undetermined* — no incorrect claim is ever made. The primary finding is the precise algebraic characterisation of this certification criterion: the *U*-th power residue partition of (ℤ/*N*ℤ)×. The undetermined set 𝒰(*p*, 𝒲) consists entirely of prime numbers — they are not composites, not exceptions, not errors. They are primes. They are also a discrete algebraic subset within ℙ, defined by a specific group-theoretic condition, distinct from the certified set yet equal members of the primes. This is the central finding that motivates the two conjectures below.

We further provide the exhaustive list of all known strictly deterministic primality certificate theorems, establishing the landscape of candidates for future inversions via the same method. Iterated prime generation is shown to be formally equivalent to Pratt certificate construction, producing certified primes at cryptographic scale — well beyond the operational range of classical sieves.

Two novel conjectures are advanced by this paper. **Conjecture 1:** the relational inversion of a deterministic primality certificate theorem is a general method for identifying and isolating discrete algebraically defined subsets within the prime numbers — sets of numbers that are fully prime, yet belong to a distinct algebraic class within ℙ — and the systematic application of this method across the known family of such theorems constitutes a road map for decomposing ℙ via its constituent algebraic layers. **Conjecture 2:** the prime numbers are not a singular, undifferentiated set but the union of multiple overlapping, algebraically distinct subsets, each characterised by a specific group-theoretic condition, whose full structure has not yet been mapped. The present paper identifies the first such subset via the Pocklington inversion and establishes the method by which further subsets may be found.