Representation of natural numbers by sum of four squares of almost-prive having a special form

TitleRepresentation of natural numbers by sum of four squares of almost-prive having a special form
Publication TypeJournal Article
Year of Publication2020
AuthorsPetrov ZH, Todorova TL
JournalAnnuaire de l’Université de Sofia “St. Kliment Ohridski”. Faculté de Mathématiques et Informatique
Volume107
Start Page15
Pagination15-29
ISSN1313-9215 (Print) 2603-5529 (Online)
Keywordsalmost-primes, Lagrange's equation, quadratic irrational numbers
Abstract

In this paper we consider the equation $x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = N$, where $N$ is a sufficiently large integer and prove that if $\eta $ is quadratic irrational number and $0 < \lambda < \frac{1}{10}$, then it has a solution in almost-prime numbers $x_{1}, \ldots , x_{4}$, such that $\{ \eta x_{i} \} < N^{-\lambda }$ for $i = 1, \ldots , 4$.

DOI10.60063/GSU.FMI.107.15-29
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