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

2025/06/18 by Artyom Radomskii, Radomskii, Artyom
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.2506.15641

arxiv created 2026/08/01 · arxiv updated 2026/08/04

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\, such that m = [(log N) (log log N)^1/325565], 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.

Citations

• Long strings of consecutive composite values of polynomials

Related

• Long strings of consecutive composite values of polynomials
• Composite polynomials in linear recurrence sequences
• A polynomial composites
• Irreducibility of polynomials with a large gap
• Polynomials Modulo Composite Numbers: Ax-Katz type theorems for the structure of their solution sets