Geodesic
An asymptotic formula for Goldbach’s conjecture with monic polynomials in $\mathbb {Z}[\theta ][x]$
Abílio Lemos1 ; Anderson Luís Albuquerque de Araujo1
1CCE, Departamento de Matemática Universidade Federal de Viçosa 36570-900, Viçosa, MG, Brasil
Colloquium Mathematicum, Tome 148 (2017) no. 2, pp. 215-223
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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Let $k\geq 2$ be a squarefree integer, and $$ \theta=\begin{cases} \sqrt{-k} \text{if }-k\not\equiv 1 \pmod4,\\ {(\sqrt{-k}+1)}/{2} \text{if }-k\equiv 1 \pmod4.\end{cases} $$ We prove that the number $R(y)$ of representations of a monic polynomial $f(x)\in \mathbb Z[\theta][x]$, of degree $d\geq 1$, as a sum of two monic irreducible polynomials $g(x)$ and $h(x)$ in $\mathbb Z[\theta][x]$, with the coefficients of $g(x)$ and $h(x)$ bounded in modulus by $y$, is asymptotic to $(4y)^{2d-2}$.
DOI : 10.4064/cm6948-7-2016
Keywords: geq squarefree integer theta begin cases sqrt k text k equiv pmod sqrt k text k equiv pmod end cases prove number representations monic polynomial mathbb theta degree geq sum monic irreducible polynomials mathbb theta coefficients bounded modulus asymptotic d

Abílio Lemos  1   ; Anderson Luís Albuquerque de Araujo  1

1 CCE, Departamento de Matemática Universidade Federal de Viçosa 36570-900, Viçosa, MG, Brasil 
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Abílio Lemos; Anderson Luís Albuquerque de Araujo. An asymptotic formula for Goldbach’s conjecture with monic polynomials in $\mathbb {Z}[\theta ][x]$. Colloquium Mathematicum, Tome 148 (2017) no. 2, pp. 215-223. doi: 10.4064/cm6948-7-2016

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