A000445 Latest possible occurrence of the first consecutive pair of n-th power residues, modulo any prime.

9, 77, 1224, 7888, 202124, 1649375

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• more
• hard

Comments

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

Examples

			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_
		

References

• 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).

Links

• 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.

Crossrefs

Cf. A000236.

Extensions

Name edited by Christopher E. Thompson, Dec 10 2019