A Monic Quartic Polynomial Generating 54 Consecutive Prime Values

This work presents a monic quartic polynomial that generates 54 consecutive prime values in canonical form (for n=0,…,53). The subsequent value is composite, establishing a run length L=54 P(n) = n4 − 106n3 + 3959n2 − 60950n + 341227 The run of 54 consecutive prime values has been independently verified in PARI/GP. The polynomial belongs to a translation orbit under integer shifts of the variable. A centered representative within the same orbit produces 28 consecutive primes in canonical form. The relationship between the canonical and centered forms is described explicitly. The document provides the full list of prime values, the composite termination value, and a structural remark on translation invariance.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-02-18
DOI
https://doi.org/10.5281/zenodo.18685002
Primary Topic
Analytic Number Theory Research
Type
article
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article

A Monic Quartic Polynomial Generating 54 Consecutive Prime Values

Paolo Borghi
Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
article

A Monic Quartic Polynomial Generating 54 Consecutive Prime Values

Paolo Borghi
article en

Abstract

This work presents a monic quartic polynomial that generates 54 consecutive prime values in canonical form (for n=0,…,53). The subsequent value is composite, establishing a run length L=54 P(n) = n4 − 106n3 + 3959n2 − 60950n + 341227 The run of 54 consecutive prime values has been independently verified in PARI/GP. The polynomial belongs to a translation orbit under integer shifts of the variable. A centered representative within the same orbit produces 28 consecutive primes in canonical form. The relationship between the canonical and centered forms is described explicitly. The document provides the full list of prime values, the composite termination value, and a structural remark on translation invariance.

Zenodo (CERN European Organization for Nuclear Research)
Openalex Percentile: Top 46%
Analytic Number Theory Research
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