An asymptotic formula for Goldbach's conjecture with monic polynomials in ℤ[θ][x]

2013/12/27 by Abílio Lemos, Lemos, Abílio, Anderson L. A. de Araújo +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1312.7295

openalex publication_date 2013/12/27 · openalex created_date 2022/09/16 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider D=ℤ[θ], where θ= √(-k) if -k\not≡ 1 (mod 4) or θ=(√(-k)+1)/(2) if -k≡ 1 (mod 4), k≥ 2 is a squarefree integer, and we proved that the number R(y) of representations of a monic polynomial f(x)∈ ℤ[θ][x], of degree d≥ 1, as a sum of two monic irreducible polynomials g(x) and h(x) in ℤ[θ][x], with the coefficients of g(x) and h(x) bounded in complex modulus by y, is asymptotic to (4y)^2d-2.

Cited by

• Problems 66 and 67 on sums of residue classes and primes of András Sárközy's collection of unsolved problems

Related

• On quantitative analogues of the Goldbach and twin prime conjectures over F_q[t]
• The size of coefficients of certain polynomials related to the Goldbach conjecture
• Primes represented by incomplete norm forms
• Additive structure of Z(.) mod m_k (squarefree) and Goldbach's Conjecture
• Sums of two squares in short intervals in polynomial rings over finite\n fields