Consecutive power residues or nonresidues
Authors:
J. R. Rabung and J. H. Jordan
Journal:
Math. Comp. 24 (1970), 737740
MSC:
Primary 10.06
MathSciNet review:
0277469
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Abstract 
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Abstract: For any positive integers and , A. Brauer [1] has shown that there exists a number such that, for any prime number , a sequence of consecutive numbers occurs in at least one thpower class modulo . For particular and , one is sometimes able to find a least bound, , before, or at which, the first member of such a sequence must appear. In this paper, we describe a method used to compute and .
 [1]
A. Brauer, "Über Sequenzen von Potenzresten," S.B. Preuss. Akad. Wiss. Phys. Math. Kl., v. 1928, pp. 916.
 [2]
W.
H. Mills, Characters with preassigned values, Canad. J. Math.
15 (1963), 169–171. MR 0156828
(28 #71)
 [3]
M.
Dunton, Bounds for pairs of cubic
residues, Proc. Amer. Math. Soc. 16 (1965), 330–332. MR 0172838
(30 #3055), http://dx.doi.org/10.1090/S00029939196501728389
 [4]
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 0162379
(28 #5578), http://dx.doi.org/10.1090/S00255718196201623792
 [5]
D.
H. Lehmer and Emma
Lehmer, On runs of residues, Proc. Amer. Math. Soc. 13 (1962), 102–106. MR 0138582
(25 #2025), http://dx.doi.org/10.1090/S00029939196201385826
 [6]
D.
H. Lehmer, Emma
Lehmer, and W.
H. Mills, Pairs of consecutive power residues, Canad. J. Math.
15 (1963), 172–177. MR 0146134
(26 #3660)
 [7]
R.
G. Bierstedt and W.
H. Mills, On the bound for a pair of consecutive
quartic residues of a prime, Proc. Amer. Math.
Soc. 14 (1963),
628–632. MR 0154843
(27 #4787), http://dx.doi.org/10.1090/S0002993919630154843X
 [8]
John
Brillhart, D.
H. Lehmer, and Emma
Lehmer, Bounds for pairs of consecutive
seventh and higher power residues, Math.
Comp. 18 (1964),
397–407. MR 0164923
(29 #2214), http://dx.doi.org/10.1090/S0025571819640164923X
 [9]
R.
L. Graham, On quadruples of consecutive
𝑘th power residues, Proc. Amer. Math.
Soc. 15 (1964),
196–197. MR 0158855
(28 #2078), http://dx.doi.org/10.1090/S00029939196401588552
 [10]
J.
H. Jordan, Pairs of consecutive power residues or
nonresidues, Canad. J. Math. 16 (1964),
310–314. MR 0161824
(28 #5028)
 [1]
 A. Brauer, "Über Sequenzen von Potenzresten," S.B. Preuss. Akad. Wiss. Phys. Math. Kl., v. 1928, pp. 916.
 [2]
 W. Mills, "Characters with preassigned values," Canad. J. Math., v. 15, 1963, pp. 169171. MR. 28 #71. MR 0156828 (28:71)
 [3]
 M. Dunton, "Bounds for pairs of cubic residues," Proc. Amer. Math. Soc., v. 16, 1965, pp. 330332. MR 30 #3055. MR 0172838 (30:3055)
 [4]
 D. Lehmer, E. Lehmer, W. Mills & J. Selfridge, "Machine proof of a theorem on cubic residues," Math. Comp., v. 16, 1962, pp. 407415. MR 28 #5578. MR 0162379 (28:5578)
 [5]
 D. Lehmer & E. Lehmer, "On runs of residues," Proc. Amer. Math. Soc., v. 13, 1962, pp. 102106. MR. 25 #2025. MR 0138582 (25:2025)
 [6]
 D. Lehmer, E. Lehmer & W. Mills, "Pairs of consecutive power residues," Canad. J. Math., v. 15, 1963, pp. 172177. MR 26 #3660. MR 0146134 (26:3660)
 [7]
 W. Mills & R. Bierstedt, "On the bound for a pair of consecutive quartic residues modulo a prime p," Proc. Amer. Math. Soc., v. 14, 1963, pp. 628632. MR 0154843 (27:4787)
 [8]
 J. Brillhart, D. Lehmer & E. Lehmer, "Bounds for pairs of consecutive seventh and higher power residues," Math. Comp., v. 18, 1964, pp. 397407. MR 29 #2214. MR 0164923 (29:2214)
 [9]
 R. L. Graham, "On quadruples of consecutive th power residues," Proc. Amer. Math. Soc., v. 15, 1964, pp. 196197. MR. 28 #2078. MR 0158855 (28:2078)
 [10]
 J. Jordan, "Pairs of consecutive power residues or nonresidues," Canad. J. Math., v. 16, 1964, pp. 310314. MR 28 #5028. MR 0161824 (28:5028)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197002774698
PII:
S 00255718(1970)02774698
Keywords:
thpower character,
thpower residues,
thpower nonresidues,
thpower class
Article copyright:
© Copyright 1970 American Mathematical Society
