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

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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 20 pairwise coprime elements g∈Z[[x]] such that both g and g+f are irreducible.

Original languageEnglish
Pages (from-to)453-461
Number of pages9
JournalJournal of Number Theory
StatePublished - Aug 2020


  • Goldbach's conjecture
  • Rings of formal power series
  • Twin-prime conjecture

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


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