TY - JOUR

T1 - Twin-prime and Goldbach theorems for Z[[x]]

AU - Paran, Elad

N1 - Publisher Copyright:
© 2020 Elsevier Inc.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/8

Y1 - 2020/8

N2 - We show that an element f in the ring Z[[x]] of formal power series over the integers is a sum of two irreducible elements in Z[[x]] if and only if the constant term of f is of the form ±pk±ql or of the form ±pk, where p,q are prime numbers and k,l are positive integers. Moreover, if f0 is of such form, then there exist 2ℵ0 pairwise coprime elements g∈Z[[x]] such that both g and g+f are irreducible.

AB - We show that an element f in the ring Z[[x]] of formal power series over the integers is a sum of two irreducible elements in Z[[x]] if and only if the constant term of f is of the form ±pk±ql or of the form ±pk, where p,q are prime numbers and k,l are positive integers. Moreover, if f0 is of such form, then there exist 2ℵ0 pairwise coprime elements g∈Z[[x]] such that both g and g+f are irreducible.

KW - Goldbach's conjecture

KW - Rings of formal power series

KW - Twin-prime conjecture

UR - http://www.scopus.com/inward/record.url?scp=85080041790&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2019.12.019

DO - 10.1016/j.jnt.2019.12.019

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AN - SCOPUS:85080041790

SN - 0022-314X

VL - 213

SP - 453

EP - 461

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -