October 20, 2025Open Access

Long strings of composite values of polynomials and a basis of order 2

Key Points

There exist two strings of consecutive positive integers yielding composite values for a polynomial.
Given a polynomial with positive leading coefficient, composite outcomes are found in two integer sets.
The study builds on previous results to demonstrate similar properties for irreducible polynomials.
The findings suggest new insights into the behavior of polynomials and their values for integers.

Abstract

We show that for any polynomial f: Z Z with positive leading coefficient and irreducible over Q, if N is large enough then there are two strings of consecutive positive integers I₁=\n₁-m, , n₁+m\ and I₂=\n₂-m, , n₂+m\, where m = (N) (N) ^1/325525, such that I₁ I₂ 1, N, N = n₁ + n₂, and f (n) is composite for any n I₁ I₂. This extends the result in 5 which showed the same result but with f (n) =n.

Read Full Paperexternally

AI से पूछें

Bookmark

View Full Paper

Cite This Study

Artyom Olegovich Radomskii (Wed,) studied this question.

synapsesocial.com/papers/68f6379bb481a140a36cf847 https://doi.org/https://doi.org/10.48550/arxiv.2506.15641

Also Consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

AI से पूछें

Bookmark

View Full Paper