Structural Classification of Prime Gaps: AMNI 2.1.1

🔢 AMNI Classification of Prime Gaps (v2.1.1)

Every prime number p_n, except for the first two (2 and 3), can be generated from the previous one using the formula:

p_n = p_{n-1} + d \quad \text{where} \quad d = 2^k \cdot m

• d: the gap between two consecutive primes
• 2^k: a power of two (with k \geq 1)
• m: an odd positive integer, which determines the complexity of the gap

This formula leads to a structural classification of prime gaps called AMNI, based on the factorization of the odd part m.

🧩 AMNI Types

Code	Definition	Description	Example
A	m = 1	Pure power of 2	3 \to 5, gap = 2
M	m = p^a	A single prime or its power	23 \to 29, gap = 6 = 2 \cdot 3
N	m = p^a \cdot q^b	Product of exactly 2 distinct primes	e.g. gap = 30 = 2 \cdot 3 \cdot 5
I	m has 3 or more distinct prime factors	Higher structural complexity	20831323 \to 20831533, gap = 210 = 2 \cdot 3 \cdot 5 \cdot 7

🔠 Subtype Notation

• A₂, A₄…: indicates the power of 2
• M₃, M₅²…: the prime or prime power in m
• N₃×5, N₇×11²…: shows the two primes in m
• I₃, I₄…: number of distinct primes in m

🔎 Try the AMNI Prime Gap Finder

Explore thousands of primes with automatic AMNI classification using the interactive tool:

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AMNI 2.1.1 Prime Classifier – Live Tool

You can:

• Search by range (up to 1 million)
• Check individual numbers
• Filter by A, M, N, or I type
• Download results in multiple formats

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