Consecutive power residues or nonresidues

Authors:
J. R. Rabung and J. H. Jordan

Journal:
Math. Comp. **24** (1970), 737-740

MSC:
Primary 10.06

MathSciNet review:
0277469

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Abstract | References | Similar Articles | Additional Information

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 th-power 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. 9-16.**[2]**W. H. Mills,*Characters with preassigned values*, Canad. J. Math.**15**(1963), 169–171. MR**0156828****[3]**M. Dunton,*Bounds for pairs of cubic residues*, Proc. Amer. Math. Soc.**16**(1965), 330–332. MR**0172838**, 10.1090/S0002-9939-1965-0172838-9**[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**, 10.1090/S0025-5718-1962-0162379-2**[5]**D. H. Lehmer and Emma Lehmer,*On runs of residues*, Proc. Amer. Math. Soc.**13**(1962), 102–106. MR**0138582**, 10.1090/S0002-9939-1962-0138582-6**[6]**D. H. Lehmer, Emma Lehmer, and W. H. Mills,*Pairs of consecutive power residues*, Canad. J. Math.**15**(1963), 172–177. MR**0146134****[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**, 10.1090/S0002-9939-1963-0154843-X**[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**, 10.1090/S0025-5718-1964-0164923-X**[9]**R. L. Graham,*On quadruples of consecutive 𝑘th power residues*, Proc. Amer. Math. Soc.**15**(1964), 196–197. MR**0158855**, 10.1090/S0002-9939-1964-0158855-2**[10]**J. H. Jordan,*Pairs of consecutive power residues or non-residues*, Canad. J. Math.**16**(1964), 310–314. MR**0161824**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1970-0277469-8

Keywords:
th-power character,
th-power residues,
th-power nonresidues,
th-power class

Article copyright:
© Copyright 1970
American Mathematical Society