Crystalline Numbers Primes Are Their Successors - N+1 is prime if and only if N is crystalline
Year: 2026
Type: article
Abstract: A new geometric primality criterion is presented, based on a new perspective in perspective: prime numbers are not found by examining the primes themselves, but by identifying the crystalline numbers that precede them. Every prime is the successor of a crystalline number. For any integer N ≥ 1, I construct its N×N binary divisibility matrix M and overlay it with its 90-degree clockwise rotation M90. If all collisions (cells where both matrices have value 1) fall exclusively on the border of the matrix, N is called "crystalline." I prove that N is crystalline if and only if N+1 is prime. The candidate prime p is never divided, never factored, never examined — its primality is determined entirely from the geometry of p−1. The proof uses the "leaps argument": two divisibility positions summing to N+1 are coprime when N+1 is prime, forcing all collisions onto the border. The property holds from N=1 with no lower bound needed. Computational verification on all 10,000 integers from N=1 to N=10,000 yields 100% accuracy with zero exceptions. An independent Excel VBA implementation confirms identical results. This work offers a different perspective: the signature of primality can also be read from p−1, from the geometry of its divisibility pattern under a single rotation." The paper includes the complete proof, the formal equivalence (N is crystalline ⟺ ∀c: gcd(N−c, c+1)=1 ⟺ N+1 is prime), visual examples from the genesis (N=1 to 7) through larger matrices, and the Python verification code.Every integer N, when processed through the divisibility matrix overlay, produces a unique geometric pattern — like a QR code for that number. The position of collisions encodes the divisibility structure of N in a visual fingerprint that is different for every number. Crystalline numbers are those whose QR code is "clean": collisions on the border only, interior empty. Their successor is always prime.
Author: Colombo, Fabio
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7147395631
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