Genealogy Theorem for Monic Prime-Generating Polynomials

A Structural Classification via Canonical Shifts

Documented by Paolo Borghi — Zenodo preprint (2026)

This page summarizes and contextualizes the preprint The Genealogy Theorem for Monic Prime-Generating Polynomials, deposited on Zenodo with DOI 10.5281/zenodo.18284699, Version 1.2 published on 18 January 2026 under the Creative Commons Attribution 4.0 International (CC-BY-4.0) License.

We formulate and prove a genealogical theorem for prime-generating monic integer polynomials evaluated on the non-negative integers. The key result states that any such polynomial with run length L > 1 admits a unique parent with run length L − 1 under a canonical shift of the argument. This induces rooted genealogical trees whose leaves correspond to structural maxima for the prime run length. The preprint provides full formal proofs, structural corollaries, and computational examples, including a monic quartic with run L = 38 and a monic cubic with run L = 31 in the canonical setting.

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The full formal proof and extended discussion are available in the PDF on Zenodo:

Overview

The work introduces a structural framework for classifying prime-generating monic integer polynomials evaluated on the non-negative integers, in terms of genealogical trees induced by a canonical shift of the argument.

The key idea is that any such polynomial with a non-trivial run of consecutive primes sits inside a finite genealogical chain, rooted at a polynomial with run length L = 1, and that this chain is canonical (i.e. uniquely determined) under a fixed shift operation. This provides a natural way to group prime-generating polynomials into “families” sharing the same structural data, such as the first composite that ends all prime runs along the chain.

Informal statement of the theorem

Let P(n) be a monic integer polynomial of degree d ≥ 2. Assume that the absolute values |P(n)| are prime for all integers n in the range 0 ≤ n ≤ L − 1, with L ≥ 2, and that P(L) is composite. The preprint proves the following:

There exists a canonical shift of the argument that associates to P a unique parent polynomial Q(n), which is again monic and prime-generating on the non-negative integers, with run length L − 1.
Iterating this construction produces a finite chain P = P_L, P_{L−1}, …, P_2, P_1, where each P_k is monic, prime-generating with run length k, and P_1 has run length L = 1 (i.e. a single prime at n = 0, followed by a composite at n = 1).
The polynomial P_1 is called the root of P. Every monic prime-generating polynomial with L > 1 admits exactly one such root, obtained by iterating the canonical parent map.

In other words, prime-generating monic polynomials on the non-negative integers naturally organize into finite genealogical trees (“sequoias”), rooted at L = 1 polynomials. Each vertex has at most one parent (under the canonical shift), and its descendants correspond to successive “extensions” of the prime run.

Structural consequences

Genealogical trees. For each root P_1, the theorem induces a finite genealogical tree consisting of all its descendants P_k with k ≥ 1. The degree and monic nature are preserved along the tree.
Structural records. Leaves of these trees (i.e. polynomials with no further extension under the canonical shift) correspond to structural maxima for the run length L within that genealogy.
Invariants along a genealogy. Empirically, polynomials lying in the same genealogical tree share certain invariants, such as the first composite value that ends the prime run (for suitable canonical choices), as well as basic algebraic data (degree, leading coefficient, discriminant under shifts, etc.).

Examples from the paper

The preprint illustrates the genealogy theorem with explicit computational examples, including:

A monic quartic with structural record L = 38 in the canonical setting, documented on the companion page Polynomial (L = 38) .
A monic cubic with canonical run length L = 31, which likewise sits at the top of its own genealogical chain in the canonical setting, see Cubic canonical record L = 31 .

In each case, the full genealogical family is obtained by repeatedly applying the canonical shift operator: starting from the structural record (maximal L in that tree) and descending step by step to the root with L = 1. For the quartic with L = 38, this yields a complete table of monic quartic polynomials with run lengths from L = 38 down to L = 1, all sharing the same first composite and forming a single genealogical chain.

Computational verification (sketch)

The theoretical statements are complemented by exhaustive computations for concrete examples. At a high level, one proceeds as follows:

Fix a monic polynomial P(n) and evaluate P(n) for n = 0, 1, 2, …, testing the primality of |P(n)| until the first composite appears. This determines the run length L.
Construct the canonical parent Q(n) via the shift prescribed by the theorem, and verify that |Q(n)| is prime exactly for n = 0, …, L − 2, with the first composite at n = L − 1.
Iterate the parent map until reaching a polynomial with L = 1. This terminal object is the root of the genealogy for P.

All computations reported in the preprint were carried out with deterministic primality tests on 64-bit integers, so that the structural status of the examples is fully reproducible.

Record summary

Title	The Genealogy Theorem for Monic Prime-Generating Polynomials
Author	Paolo Borghi
Type	Theoretical / structural result
Focus	Genealogical trees of monic prime-generating polynomials
Domain	Consecutive primes for integer arguments n ≥ 0
Key concept	Canonical parent map and L-chains from L = 1 to structural records
Repository	Zenodo record 18284699
Version	1.2 (published 18 January 2026)
License	Creative Commons Attribution 4.0 International (CC-BY-4.0)

Zenodo reference and DOI

Official reference:

Borghi, P. (2026). The Genealogy Theorem for Monic Prime-Generating Polynomials. Zenodo. DOI: 10.5281/zenodo.18284699

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How to cite

Borghi, P. (2026). The Genealogy Theorem for Monic
Prime-Generating Polynomials. Zenodo.
https://doi.org/10.5281/zenodo.18284699

License

This work is licensed under the Creative Commons Attribution 4.0 International (CC-BY-4.0) License. You are free to share and adapt the material, provided that appropriate credit is given.

Author

Paolo Borghi
Independent researcher in number theory and prime-generating polynomials.
paolo.borghi@gmail.com

Related records

This page presents a structural overview of the genealogy theorem and its role in organizing prime-generating monic polynomials into canonical genealogical trees. For full proofs, precise definitions, and detailed examples, please refer to the Zenodo preprint.