Consecutive power residues or nonresidues
HTML articles powered by AMS MathViewer
 by J. R. Rabung and J. H. Jordan PDF
 Math. Comp. 24 (1970), 737740 Request permission
Abstract:
For any positive integers $k$ and $l$, A. Brauer [1] has shown that there exists a number $z(k,l)$ such that, for any prime number $p > z(k,l)$, a sequence of $l$ consecutive numbers occurs in at least one $k$thpower class modulo $p$. For particular $k$ and $l$, one is sometimes able to find a least bound, $\Lambda *(k,l)$, before, or at which, the first member of such a sequence must appear. In this paper, we describe a method used to compute $\Lambda *(8,2)$ and $\Lambda *(3,3)$.References

A. Brauer, "Γber Sequenzen von Potenzresten," S.B. Preuss. Akad. Wiss. Phys. Math. Kl., v. 1928, pp. 9β16.
 W. H. Mills, Characters with preassigned values, Canadian J. Math. 15 (1963), 169β171. MR 156828, DOI 10.4153/CJM19630193
 M. Dunton, Bounds for pairs of cubic residues, Proc. Amer. Math. Soc. 16 (1965), 330β332. MR 172838, DOI 10.1090/S00029939196501728389
 D. H. Lehmer, E. Lehmer, W. H. Mills, and J. L. Selfridge, Machine proof of a theorem on cubic residues, Math. Comp. 16 (1962), 407β415. MR 162379, DOI 10.1090/S00255718196201623792
 D. H. Lehmer and Emma Lehmer, On runs of residues, Proc. Amer. Math. Soc. 13 (1962), 102β106. MR 138582, DOI 10.1090/S00029939196201385826
 D. H. Lehmer, Emma Lehmer, and W. H. Mills, Pairs of consecutive power residues, Canadian J. Math. 15 (1963), 172β177. MR 146134, DOI 10.4153/CJM19630204
 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), 628β632. MR 154843, DOI 10.1090/S0002993919630154843X
 John Brillhart, D. H. Lehmer, and Emma Lehmer, Bounds for pairs of consecutive seventh and higher power residues, Math. Comp. 18 (1964), 397β407. MR 164923, DOI 10.1090/S0025571819640164923X
 R. L. Graham, On quadruples of consecutive $k$th power residues, Proc. Amer. Math. Soc. 15 (1964), 196β197. MR 158855, DOI 10.1090/S00029939196401588552
 J. H. Jordan, Pairs of consecutive power residues or nonresidues, Canadian J. Math. 16 (1964), 310β314. MR 161824, DOI 10.4153/CJM19640306
Additional Information
 © Copyright 1970 American Mathematical Society
 Journal: Math. Comp. 24 (1970), 737740
 MSC: Primary 10.06
 DOI: https://doi.org/10.1090/S00255718197002774698
 MathSciNet review: 0277469