A note on Dirichlet characters
HTML articles powered by AMS MathViewer
 by Richard H. Hudson PDF
 Math. Comp. 27 (1973), 973975 Request permission
Abstract:
Denoting by $r(k,m,p)$ the first occurrence of m consecutive kth power residues of a prime $p \equiv 1 \pmod k$, we show that $r(k,m,p) > c\log p$ for infinitely many p (c is an absolute constant) provided that k is even and $m \geqq 3$.References

A. Brauer, "Ueber Sequenzen von Potenzresten," S.B. Preuss. Akad. Wiss. Phys. Math. Kl., 1928, pp. 916.
 D. A. Burgess, A note on the distribution of residues and nonresidues, J. London Math. Soc. 38 (1963), 253–256. MR 148628, DOI 10.1112/jlms/s138.1.253
 P. D. T. A. Elliott, Some notes on $k$th power residues, Acta Arith. 14 (1967/68), 153–162. MR 228419, DOI 10.4064/aa142153162
 Richard H. Hudson, A bound for the first occurrence of three consecutive integers with equal quadratic character, Duke Math. J. 40 (1973), 33–39. MR 314743
 D. H. Lehmer and Emma Lehmer, On runs of residues, Proc. Amer. Math. Soc. 13 (1962), 102–106. MR 138582, DOI 10.1090/S00029939196201385826
 D. H. Lehmer, E. Lehmer, W. H. Mills, and J. L. Selfridge, Machine proof of a theorem on cubic residues, Math. Comp. 16 (1962), 407–415. MR 162379, DOI 10.1090/S00255718196201623792
 U. V. Linnik, On the least prime in an arithmetic progression. I. The basic theorem, Rec. Math. [Mat. Sbornik] N.S. 15(57) (1944), 139–178 (English, with Russian summary). MR 0012111
 W. H. Mills, Bounded consecutive residues and related problems, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 170–174. MR 0176958
 J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
 Hans Salié, Über den kleinsten positiven quadratischen Nichtrest nach einer Primzahl, Math. Nachr. 3 (1949), 7–8 (German). MR 33844, DOI 10.1002/mana.19490030104 I. Schur, "Multiplikativ signierte Folgen postiver ganzer Zahlen," Gesammelte Abhandlungen von I. Schur, Springer, Berlin, 1973.
Additional Information
 © Copyright 1973 American Mathematical Society
 Journal: Math. Comp. 27 (1973), 973975
 MSC: Primary 10H35; Secondary 10H15
 DOI: https://doi.org/10.1090/S00255718197303378508
 MathSciNet review: 0337850