Author: Bruce Hunter Jr.
Date: 2025-05-30
@BruceHunterJr
This paper introduces the Hunter Equation, a novel affine update rule I derived through the exploration of number theory, specifically the structure of base‑10 digits and the properties of the first two prime numbers. The equation models directional, sign‑dependent transitions in a compact and elegant form:
$$ P_2 \;=\; \frac{3P_1 \;+\;\sigma\,x}{2}, \quad \sigma \in \{-1, +1\}, \quad x \in \mathbb{N}_0 $$
Discovered independently in 2018, this equation captures a minimal, symbolic representation of state transitions influenced by discrete polarity. Applications range from control systems to probabilistic models and binary decision logic. Most notably, the equation reveals a striking prime hunting property — the ability to generate a consecutive prime sequence by selecting appropriate \(\sigma\) and \(x\).
I discovered the Hunter Equation through personal exploration of number theory. As a self‑taught independent thinker and Aspy — shorthand for self‑identified on the autism spectrum, I began investigating the structure of the natural digits 0, 1, ..., 9 and the unique properties of the first two primes: 2 (even) and 3 (odd) in 2018. From these foundational elements, I observed patterns that led to a consistent linear‑affine structure with a directional switch.
This work was conducted without assistance from artificial intelligence or formal institutional backing. In fact, I do not hold a college degree—my efforts stem purely from a deep personal interest in prime numbers and a lifelong determination to uncover patterns that may contribute meaningfully to one of the most challenging problems in mathematics: the search for a formal proof of the Riemann Hypothesis. This personal mission is the foundation for the symbolic mechanisms developed in this work, and directly motivates the potential research directions explored in Section 7. I’m releasing what I have found thus far—what I call the Hunter Equation—to the world so that it is not lost in one of my notebooks as an unreleased finding, especially if I were to leave this world. I didn’t want my efforts to vanish without being shared. This formal documentation ensures that the equation and its properties are preserved for others to study, apply, and potentially build upon. All I ask is that anyone who uses or expands upon this work gives proper credit.
The refined form of the equation is:
$$ P_2 \;=\; \frac{3P_1 + \sigma x}{2}, \quad \sigma \in \{-1, +1\}, \quad x \in \mathbb{N}_0 $$
(2.1)
Where:
This compact affine form captures both deterministic structure and symbolic flexibility. The sign and size of \(x\) allow discrete control of directional state updates.
From analysis of the digits 0 through 9 and primes 2 and 3, I noticed primitive structural rules governing parity and transition. The even‑odd polarity seeded the directional transformation logic. When expanded, the form naturally revealed linear‑affine behavior, modulated by a discrete signed input.
This equation possesses a remarkable prime hunting property. Starting from \(P_{1}=1\), and feeding \(P_{2}\) back as the next \(P_{1}\), one can reach each of the first prime numbers—in order—by choosing the correct \(\sigma\) and smallest whole number \(x\).
Note: Although \(1\) is not prime, we use \(P_{1}=1\) as a seed. To reach \(2\), solve \(\frac{3\cdot1 + \sigma\,x}{2} = 2\), which forces \(\sigma = +1\) and \(x = 1\).
| Step | \(P_{1}\) | \(x\) | \(\sigma\) | \(P_{2}\) |
|---|---|---|---|---|
| 1 | 1 | 1 | + | 2 |
| 2 | 2 | 0 | + | 3 |
| 3 | 3 | 1 | + | 5 |
| 4 | 5 | 1 | − | 7 |
| 5 | 7 | 1 | + | 11 |
| 6 | 11 | 7 | − | 13 |
| 7 | 13 | 5 | − | 17 |
| 8 | 17 | 13 | − | 19 |
| 9 | 19 | 11 | − | 23 |
| 10 | 23 | 11 | − | 29 |
| 11 | 29 | 25 | − | 31 |
While the Hunter Equation is not a formal proof of the Riemann Hypothesis (RH), it introduces a symbolic and structural method for tracing the prime sequence. If the equation reliably produces every prime via a deterministic rule, it could provide a discrete analog of the behavior captured by the zeta function’s analytic continuation.
In particular, the recursive mechanism mimics a discrete flow through prime space — something that traditional continuous analysis approximates with integrals and complex zeros.
Potential research directions include:
If a link between this recursive behavior and the analytic structure of primes can be formalized — especially in how it models or bounds the error in \(\pi(x)\) or Chebyshev functions — it may inform progress toward or alongside a proof of RH.
The Hunter Equation is a symbolic and functional expression of directional state update dynamics, derived from independent reasoning grounded in number theory. It encodes signed control over affine transformation, and surprisingly, also demonstrates a prime-generating structure when initialized properly.
This opens future work on deterministic models of prime behavior, dynamic symbolic computation, and deeper explorations of algebraic recursion in discrete mathematics. In particular, the connection between prime gaps and mod‑6 residue structures hints that the equation reflects deep structure in the distribution of primes.
As the author, I will continue developing this work, specifically along the research directions proposed in Section 7. The goal is to explore whether the Hunter Equation can yield new insights into the zeta function, prime counting functions, or even contribute toward a proof of the Riemann Hypothesis.
This work was independently discovered by me, Bruce Hunter Jr., without institutional affiliation or AI assistance. AI was used for typesetting assistance only; all mathematical insights are my own. I want to acknowledge the profound influence of Bernhard Riemann — without his visionary insights and the enduring mystery of the Riemann Hypothesis, I would not have embarked on this mathematical journey. I have spent years studying and developing these ideas, often with nothing more than a pencil and a notebook, driven by passion and curiosity. This work represents my attempt to preserve and share the results of those many solitary nights of exploration.
For those interested in discussing or building on this work, I can be contacted via X (formerly Twitter) at @BruceHunterJr.