What question did this study set out to answer?

The aim is to establish a Genealogical Theorem for monic polynomials that generate primes.

January 20, 2026Open Access

The Genealogy Theorem for Monic Prime‑Generating Polynomials

Key Points

The aim is to establish a Genealogical Theorem for monic polynomials that generate primes.
Proposed and proved a Genealogical Theorem for prime-generating monic integer polynomials.
Analyzed properties of polynomials evaluated on non-negative integers.
Developed rooted genealogical trees based on polynomial run lengths.
Demonstrated the existence of a unique parent polynomial for those with run length greater than one.
Provided examples of specific monic polynomials with prime run lengths, including one with L = 38.

Abstract

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 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. We provide 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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