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 \(1\le x,y\le H\), Dimitrov established an asymptotic formula for the number of the square-free values attained by the polynomial \(f\left( x,y\right) =\left( x^2+y^2+1\right) \left( x^2+y^2+2\right) \). Motived by the work of Dimitrov, in this paper, we give an asymptotic formula for the consecutive square-free numbers \(x_1^2+\cdots +x_k^2+1\), \(x_1^2+\cdots +x_k^2+2\).
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Communicated by B. Sury.
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Chen, B. On the consecutive square-free values of the polynomials \(x_1^2+\cdots +x_k^2+1\), \(x_1^2+\cdots +x_k^2+2\). Indian J Pure Appl Math 54, 743–756 (2023). https://doi.org/10.1007/s13226-022-00292-z
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DOI: https://doi.org/10.1007/s13226-022-00292-z