This page summarizes and contextualizes the preprint Genealogical Dynamics of Monic Prime-Generating Polynomials, deposited on Zenodo.
The work introduces a genealogical structure for monic polynomials with integer coefficients generating consecutive prime numbers on the non-negative integers. The central idea is that the canonical translation
\[ f(n)\mapsto f(n-1) \]
induces a highly rigid discrete dynamical structure on the length of the prime sequences generated by the polynomial.
As a consequence, every polynomial genealogy is finite and bounded by two collapse states. This structure naturally induces a discrete forest of isolated genealogical trees.
The full formal proof and extended discussion are available in the PDF on Zenodo:
Prime-generating polynomials are usually studied as isolated objects: a given polynomial is tested on the sequence \(n=0,1,2,\dots\), and the number of consecutive prime values is recorded.
This work proposes a different perspective. A prime-generating monic polynomial is not viewed as a single isolated object, but as an element of a discrete genealogical structure generated by translations of the argument.
The fundamental invariant is the prime run length \(L(f)\), defined as the maximum number of consecutive positive prime values produced by \(f(n)\) starting from \(n=0\).
For a polynomial \(f(n)\in\mathbb Z[n]\), the prime run length \(L(f)\) is defined by
\[ L(f):= \max \left\{ k\ge0 \;\middle|\; f(n)\text{ is a positive prime number for every } n=0,1,\dots,k-1 \right\}. \]
If \(f(0)\) is not a positive prime number, then \(L(f)=0\). Throughout the work, only positive primes are considered; no absolute values are used.
Given a canonical monic polynomial \(f(n)\), its genealogical child is defined as
\[ g(n)=f(n-1). \]
The genealogical parent is defined as
\[ p(n)=f(n+1). \]
These two translations preserve monicity, degree, and integrality of the coefficients.
The shift \(f(n)\mapsto f(n-1)\) introduces exactly one new value at the beginning of the sequence:
\[ g(0)=f(-1). \]
All the other values
\[ g(1),g(2),\dots,g(L) \]
correspond to the already prime values
\[ f(0),f(1),\dots,f(L-1). \]
Therefore, if \(f(-1)\) is a positive prime, the run length increases by one. If \(f(-1)\) is not a positive prime, the new polynomial fails immediately at \(n=0\), and its run length collapses to \(0\).
In this framework, a genealogical tree is a finite linear sequence of canonical monic polynomials
\[ f_1,f_2,\dots,f_m \]
such that
\[ f_{k+1}(n)=f_k(n-1), \qquad L(f_{k+1})=L(f_k)+1. \]
Every genealogical tree is bounded by two collapse states:
\[ L=0 \to L=1 \to L=2 \to \cdots \to L_{\max} \to L=0. \]
The set of canonical monic polynomials naturally organizes itself into a discrete collection of finite isolated genealogical trees.
The prime run length \(L\) therefore appears not as an isolated property of the polynomial, but as a dynamical coordinate along a genealogical translation line.
Consider the monic cubic polynomial
\[ C_{31}(n)=n^3+54n^2-2329n+20507. \]
It generates consecutive positive prime numbers for
\[ n=0,1,\dots,30. \]
Therefore,
\[ L(C_{31})=31. \]
\[ 0 \to 1 \to 2 \to 3 \to \cdots \to 31 \to 0 \]
| Level | Polynomial |
|---|---|
| \(L=31\) | \(n^3+54n^2-2329n+20507\) |
| \(L=30\) | \(n^3+57n^2-2218n+18233\) |
| \(L=29\) | \(n^3+60n^2-2101n+16073\) |
| \(L=28\) | \(n^3+63n^2-1978n+14109\) |
| \(L=27\) | \(n^3+66n^2-1849n+12323\) |
| \(L=26\) | \(n^3+69n^2-1714n+10697\) |
| \(L=25\) | \(n^3+72n^2-1573n+9213\) |
| \(L=24\) | \(n^3+75n^2-1426n+7853\) |
| \(L=23\) | \(n^3+78n^2-1273n+6599\) |
| \(L=22\) | \(n^3+81n^2-1114n+5433\) |
| \(L=21\) | \(n^3+84n^2-949n+4337\) |
| \(L=20\) | \(n^3+87n^2-778n+3293\) |
| \(L=19\) | \(n^3+90n^2-601n+2283\) |
| \(L=18\) | \(n^3+93n^2-418n+1289\) |
| \(L=17\) | \(n^3+96n^2-229n+293\) |
| \(L=16\) | \(n^3+99n^2-34n-723\) |
| \(L=15\) | \(n^3+102n^2+167n-1777\) |
| \(L=14\) | \(n^3+105n^2+374n-2887\) |
| \(L=13\) | \(n^3+108n^2+587n-4071\) |
| \(L=12\) | \(n^3+111n^2+806n-5347\) |
| \(L=11\) | \(n^3+114n^2+1031n-6733\) |
| \(L=10\) | \(n^3+117n^2+1262n-8247\) |
| \(L=9\) | \(n^3+120n^2+1499n-9907\) |
| \(L=8\) | \(n^3+123n^2+1742n-11731\) |
| \(L=7\) | \(n^3+126n^2+1991n-13737\) |
| \(L=6\) | \(n^3+129n^2+2246n-15943\) |
| \(L=5\) | \(n^3+132n^2+2507n-18367\) |
| \(L=4\) | \(n^3+135n^2+2774n-21027\) |
| \(L=3\) | \(n^3+138n^2+3047n-23941\) |
| \(L=2\) | \(n^3+141n^2+3326n-27127\) |
| \(L=1\) | \(n^3+144n^2+3611n-30603\) |
| Title | Genealogical Dynamics of Monic Prime-Generating Polynomials |
|---|---|
| Author | Paolo Borghi |
| Type | Theoretical / computational preprint |
| Focus | Genealogical dynamics of monic prime-generating polynomials |
| Domain | Consecutive positive primes for integer arguments \(n\ge0\) |
| Key concept | The \(L+1/0\) bifurcation under the child map \(f(n)\mapsto f(n-1)\) |
| Example | Complete genealogy of a monic cubic with \(L=31\) |
| Repository | Zenodo record 20415444 |
| License | Creative Commons Attribution 4.0 International (CC-BY-4.0) |
Official reference:
Borghi, P. (2026). Genealogical Dynamics of Monic Prime-Generating Polynomials. Zenodo. DOI: 10.5281/zenodo.20415444
Borghi, P. (2026). Genealogical Dynamics of Monic Prime-Generating Polynomials. Zenodo. https://doi.org/10.5281/zenodo.20415444
This work is licensed under the Creative Commons Attribution 4.0 International (CC-BY-4.0) License.
Paolo Borghi
Independent researcher in number theory and prime-generating polynomials.
This page presents a structural overview of the genealogical dynamics of monic prime-generating polynomials. For complete proofs, precise definitions, and the full formal exposition, please refer to the Zenodo preprint.