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

 Title Representation of natural numbers by sum of four squares of almost-prive having a special form Publication Type Journal Article Year of Publication 2020 Authors Petrov ZH, Todorova TL Journal Annuaire de l’Université de Sofia “St. Kliment Ohridski”. Faculté de Mathématiques et Informatique Volume 107 Start Page 15 Pagination 15-29 ISSN 1313-9215 (Print) 2603-5529 (Online) Keywords almost-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 $i = 1, \ldots , 4$.
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