On the consecutive square-free values of the polynomials $$x_1^2+\cdots +x_k^2+1$$ x 1 2 + ⋯ + x k 2 + 1, $$x_1^2+\cdots +x_k^2+2$$ x 1 2 + ⋯ + x k 2 + 2

Bo Chen ()
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Bo Chen: Sanda University

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 3, 743-756

Abstract: Abstract The problem of evaluating square-free values of polynomials is a classical problem that has attracted many authors, including Eetermann, Carlitz, Tolev and Zhou. Recently, for $$1\le x,y\le H$$ 1 ≤ x , y ≤ H , Dimitrov established an asymptotic formula for the number of the square-free values attained by the polynomial $$f\left( x,y\right) =\left( x^2+y^2+1\right) \left( x^2+y^2+2\right) $$ f x , y = x 2 + y 2 + 1 x 2 + y 2 + 2 . Motived by the work of Dimitrov, in this paper, we give an asymptotic formula for the consecutive square-free numbers $$x_1^2+\cdots +x_k^2+1$$ x 1 2 + ⋯ + x k 2 + 1 , $$x_1^2+\cdots +x_k^2+2$$ x 1 2 + ⋯ + x k 2 + 2 .

Keywords: square-free numbers; polynomial; asymptotic formula; 11L05; 11N37 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00292-z

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