Hello everyone,
I would like to share a short preprint that I am publishing here in Reddit (`r/numbertheory`) regarding a structural variant of explicit Willans-type formulas for generating the $n$-th prime number ($p_n \ge 5$).
### Abstract
This paper presents a structural variant of explicit Willans-type formulas for generating the $n$-th prime number $p_n$. Through the integration of an arithmetic filtering kernel based on floor function differences, an indexed accumulation operator, and an outer linear transformation, a deterministic analytical expression is established that accurately maps the sequence of prime numbers in the domain $p_n \ge 5$.
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### The Three Fundamental Formulas of the System
1. **Formula 1: The Arithmetic Filtering Kernel**
We define the indicator function $f(i)$ via floor differences, designed to evaluate and isolate valid indices while excluding composite numbers:
$$f(i) = \left\lfloor \frac{1}{1 + \sum_{k=1}^{\lfloor\frac{i-1}{3}\rfloor} \left(\lfloor\frac{i-k}{2k+1}\rfloor - \lfloor\frac{i-k-1}{2k+1}\rfloor\right)} \right\rfloor$$
2. **Formula 2: The Accumulated Function $Z_n$**
To count and govern the growth until reaching the $n$-th term, we define the accumulated operator $Z_n$ using an explicit upper bound:
$$Z_n = 1 + \sum_{j=1}^{2n+1} \left\lfloor \left( \frac{n}{\sum_{k=1}^{j} \left\lfloor \frac{1}{1 + \sum_{m=1}^{\lfloor\frac{k-1}{3}\rfloor} \left(\lfloor\frac{k-m}{2m+1}\rfloor - \lfloor\frac{k-m-1}{2m+1}\rfloor\right)} \right\rfloor} \right)^{1/n} \right\rfloor$$
3. **Formula 3: The Final Expression for the $n$-th Prime ($p_n$)**
From the previous building blocks, the $n$-th prime number $p_n$ is obtained by applying the outer linear transformation with upper limit $M$:
$$p_n = 3 + 2 \sum_{j=1}^{M} \left\lfloor \left( \frac{n}{\sum_{i=1}^{j} \left\lfloor \frac{1}{1 + \sum_{k=1}^{\lfloor\frac{i-1}{3}\rfloor} \left(\lfloor\frac{i-k}{2k+1}\rfloor - \lfloor\frac{i-k-1}{2k+1}\rfloor\right)} \right\rfloor} \right)^{1/n} \right\rfloor$$
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### Domain Analysis and Results
* **Initial behavior ($n = 1$):** Evaluating the structure for $n = 1$, the accumulator yields a base value, resulting in $p_1 = 3 + 2(1) = 5$.
* **Generated sequence:** The formula deterministically and strictly maps the sequence of prime numbers greater than or equal to 5 ($5, 7, 11, 13, 17, \dots$).
**Full Preprint & DOI:**
You can access the full paper on Zenodo:
* **DOI:** `10.5281/zenodo.23250898`
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### Questions for the Community:
1. **Differences with classical formulas:** As you may notice in this formulation shared here, it is different from the classical Willans formula (which uses factorials and trigonometric/cosine functions); instead, it uses a purely arithmetic kernel based on floor function differences. Do you know of other works exploring similar filtering structures without resorting to factorials or trigonometry?
2. **Computational Scaling:** Since explicit formulas involving nested floor functions and roots grow rapidly in computational complexity for large $n$, has anyone experimented with optimizing similar algebraic structures in Python or SageMath?
I would love to hear the community's thoughts, suggestions, or feedback on this formulation!
