Modulo-3 Complementarity and Prime-Value Conjectures for p³ ± 2 over Prime Inputs
Year: 2026
Type: article
Abstract: For prime inputs p > 3, the values p³ − 2 and p³ + 2 obey a complementary divisibility rule modulo 3. If p ≡ 1 (mod 3), then p³ + 2 is divisible by 3; if p ≡ 2 (mod 3), then p³ − 2 is divisible by 3. In each case, the expression with the opposite sign is eligible, but not guaranteed, to be prime. This paper states two conjectures: infinitely many primes p ≡ 1 (mod 3) produce prime values of p³ − 2, and infinitely many primes p ≡ 2 (mod 3) produce prime values of p³ + 2. The recorded minus-lane computation covers 6,110,100 consecutive prime inputs through p = 106,431,659. During the final 5,000,000-input stage, 166,507 prime outputs were observed, compared with a reported heuristic expectation of 166,343.25. These finite computations are consistent with a Bateman–Horn and local-sieve interpretation but do not prove either infinitude conjecture. No separately audited quantitative plus-lane result is claimed in this version. The accompanying reproduction package contains source code, pinned dependencies, computational results, benchmark output, metadata, and SHA-256 checksums intended to support independent reproduction and auditing.
Author: Schimmel, Kevin Mark
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208813027
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