Abstract
Bounds for the second and third smallestprime k-th power nonresidues of odd primes p have been given by Alfred Brauer, Clifton Whyburn, and L. K. Hua. Bounds for the n-th prime residue, n≥4, do not appear in the literature and it would be difficult to obtain bounds as sharp as p1/4 if n is large and k is small. In this note we use the character sum estimates of D. A. Burgess to show that there are on the order of log p/log log pprime k-th power nonresidues less than p1/4 +∈ for every ∈>0 and sufficiently large p.
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References
Alfred Brauer, Ueber den kleinsten quadratischen, Math.Zeitschr. 33 (1931), 161–176
—, On the non-existence of the Euclidean algorithm in certain quadratic number fields, Amer. J. Math. 62 (1940), 697–716
D. A. Burgess, On character sums and primitive roots, Proc. London Math. Soc. 12 (1962), 179–192
L. K. Hua, On the distribution of quadratic non-residues and the Euclidean algorithm in real quadratic fields, I, Trans. Amer. Math. Soc. 56 (1944), 537–546
Richard H. Hudson, Prime k-th power non-residues, Acta Arith. 23 (1973), 89–106
Clifton Whyburn, The second smallest quadratic non-residue, Duke Math. J. 32 (1965), 519–528
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Hudson, R.H. A note on prime k-th power nonresidues. Manuscripta Math 42, 285–288 (1983). https://doi.org/10.1007/BF01169590
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DOI: https://doi.org/10.1007/BF01169590