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A000445 Latest possible occurrence of the first consecutive pair of n-th power residues, modulo any prime.
(Formerly M4652 N1991)
9, 77, 1224, 7888, 202124, 1649375 (list; graph; refs; listen; history; text; internal format)
The paper by Adolf Hildebrand proves that a(n) is finite for all n. - Christopher E. Thompson, Dec 05 2019
Don Reble has reported computations proving that 1499876 <= a(8) <= 1508324, which improves on the references below. Note also that it shows a(8) < a(7). - Christopher E. Thompson, Jan 14 2020
P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory. L'Enseignement Math., Geneva, 1980, p. 87.
W. H. Mills, Bounded consecutive residues and related problems, pp. 170-174 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. G. Bierstedt, W. H. Mills, On the bound for a pair of consecutive quartic residues of a prime, Proc. Amer. Math. Soc. 14, 628-632 (1963).
J. Brillhart, D. H. Lehmer and E. Lehmer, Bounds for pairs of consecutive seventh and higher power residues, Math. Comp. 18 (1964), 397-407.
M. Dunton, Bounds for Pairs of Cubic Residues, Proc. Amer. Math. Soc. 16 (1965), 330-332.
Adolf Hildebrand, On consecutive k-th power residues. II., Michigan Math. J., 38 (1991), no. 2, 241--253.
J. H. Jordan, Pairs of consecutive power residues or non-residues, Canad. J. Math., 16 (1964), 310-314.
J. R. Rabung and J. H. Jordan, Consecutive power residues or nonresidues, Math. Comp. 24 (1970), 737-740.
Don Reble, More terms for A000445?, posting to SeqFan mailing list, Dec 19 2019.
Every large prime has a pair of consecutive quadratic (n=2) residues which appear not later than 9,10, so a(2)=9. - Len Smiley
Cf. A000236.
Sequence in context: A355372 A046150 A124131 * A046196 A231596 A350428
Name edited by Christopher E. Thompson, Dec 10 2019

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