Power nonresidues modulo a prime number in a special greatest integer-sequence
The author obtains an estimate for the least quadratic nonresidue modulo a prime number in the greatest integer-sequence \([\alpha n]\), where \(\alpha\) is a real irrational number. This subject originates from the works of Vinogradov. Later many important interesting results were obtained by Davenport and Erdős. D. A. Burgess proved the theorem: For any \(\varepsilon >0\) the least quadratic nonresidue modulo \(p\) does not exceed \(D(\varepsilon) p^{\frac{1}{4\sqrt e}+\varepsilon}\). Let \(n_{\min}\) denote the least \(n\) such that the corresponding number \([\alpha n]\bmod p\) from the greatest integer-sequence is a quadratic nonresidue modulo \(p\). Here, the author proves the following: Theorem. Let \(\alpha\) be a real irrational number and assume that the incomplete quotients of the continued fraction expansion of \(\alpha\) are bounded. Then \(n_{\min}\leq Cp^{\frac{1} {2\sqrt e}+\varepsilon}\), where \(C=C (\alpha,\varepsilon)\) is a constant depending on \(\alpha\) and \(\varepsilon\) only, and \(\varepsilon>0\) is an arbitrarily small constant.
scientific article; zbMATH DE number 1899987