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%I M4652 N1991 #53 Jan 02 2023 12:30:46
%S 9,77,1224,7888,202124,1649375
%N Latest possible occurrence of the first consecutive pair of n-th power residues, modulo any prime.
%C The paper by Adolf Hildebrand proves that a(n) is finite for all n. - _Christopher E. Thompson_, Dec 05 2019
%C _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
%D P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory. L'Enseignement Math., Geneva, 1980, p. 87.
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H R. G. Bierstedt, W. H. Mills, <a href="https://doi.org/10.1090/S0002-9939-1963-0154843-X">On the bound for a pair of consecutive quartic residues of a prime</a>, Proc. Amer. Math. Soc. 14, 628-632 (1963).
%H J. Brillhart, D. H. Lehmer and E. Lehmer, <a href="http://dx.doi.org/10.1090/S0025-5718-1964-0164923-X">Bounds for pairs of consecutive seventh and higher power residues</a>, Math. Comp. 18 (1964), 397-407.
%H M. Dunton, <a href="https://doi.org/10.1090/S0002-9939-1965-0172838-9">Bounds for Pairs of Cubic Residues</a>, Proc. Amer. Math. Soc. 16 (1965), 330-332.
%H Adolf Hildebrand, <a href="https://doi.org/10.1307/mmj/1029004331">On consecutive k-th power residues. II.</a>, Michigan Math. J., 38 (1991), no. 2, 241--253.
%H J. H. Jordan, <a href="http://dx.doi.org/10.4153/CJM-1964-030-6">Pairs of consecutive power residues or non-residues</a>, Canad. J. Math., 16 (1964), 310-314.
%H J. R. Rabung and J. H. Jordan, <a href="https://doi.org/10.1090/S0025-5718-1970-0277469-8">Consecutive power residues or nonresidues</a>, Math. Comp. 24 (1970), 737-740.
%H Don Reble, <a href="http://list.seqfan.eu/oldermail/seqfan/2019-December/020300.html">More terms for A000445?</a>, posting to SeqFan mailing list, Dec 19 2019.
%e 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_
%Y Cf. A000236.
%K nonn,nice,more,hard
%O 2,1
%A _N. J. A. Sloane_
%E Name edited by _Christopher E. Thompson_, Dec 10 2019
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