2025/06/26 by Jean‐Marc Deshouillers, Deshouillers, Jean-Marc, Sunil Naik +1
Mathematics · #11B05 #11B25 11B50 #11K31 #11N56 #41A58 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2506.20956
openalex publication_date 2025/06/26 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28
The investigation of primes in certain arithmetic sequences is one of the fundamental problems in number theory and especially, finding blocks of distinct primes has gained a lot of attention in recent years. In this context, we prove the existence of long blocks of k-wise coprime elements in certain regular sequences. More precisely, we prove that for any positive integers H ≥ k ≥ 2 and for a real-valued k-times continuously differentiable function f ∈ Ck( [1, ∞)) satisfying limx → ∞ f(k)(x) = 0 and \limsupx → ∞ f(k-1)(x) = ∞, there exist infinitely many positive integers n such that gcd( \lfloor f(n+i1)\rfloor, \lfloor f(n+i2)\rfloor, ⋯, \lfloor f(n+ik)\rfloor ) ~=~ 1 for any integers 1 ≤ i1 < i2 < ⋯ < ik ≤ H. Further, we show that there exists a subset A ⊆ ℕ having upper Banach density one such that gcd(\lfloor f(n1) \rfloor, \lfloor f(n2) \rfloor, ⋯, \lfloor f(nk) \rfloor) ~=~ 1 for any distinct integers n1, n2, ⋯, nk ∈ A.