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A330274 Largest positive x such that (x,x+n) is the smallest pair of quadratic residues with difference n, modulo any prime. 1
9, 4, 1, 10, 4, 14, 9, 1, 9, 12, 5, 4, 11, 13, 1, 9, 10, 15, 11, 10, 4, 14, 4, 1, 15, 10, 9, 26, 16, 12, 9, 4, 16, 21, 1, 21, 23, 14, 16, 9, 15, 14, 17, 16, 4, 22, 9, 1, 16, 25, 25, 29, 19, 16, 9, 25, 30, 27, 16, 4, 24, 22, 1, 21, 16, 22, 29, 22, 31, 30, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

There is a finite limit for any n. By considering the pairs (1,n+1), (n^2,n^2+n), (n,2n), (4n,5n), (9n,10n) it can be seen that a(n) <= max(9n,n^2).

REFERENCES

Richard K. Guy, Unsolved Problems in Number Theory, Springer-Verlag (1981,1994,2004), section F6 "Patterns of quadratic residues".

LINKS

Christopher E. Thompson, Table of n, a(n) for n = 1..1000

Emma Lehmer, Patterns of power residues, J. Number Theory 17 (1983) 37-46.

EXAMPLE

If each of the pairs (1,5),(4,8),(6,10),(3,7) are not both quadratic residues, then (10,14) must be. Moreover, if 3 is a quadratic residue but 2,5,7 and 13 are not, then (10,14) is the smallest pair (x,x+4) which are both quadratic residues. Therefore, a(4)=10.

CROSSREFS

Sequence in context: A021519 A199780 A292684 * A248197 A199291 A091661

Adjacent sequences:  A330271 A330272 A330273 * A330275 A330276 A330277

KEYWORD

nonn

AUTHOR

Christopher E. Thompson, Dec 08 2019

STATUS

approved

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Last modified July 26 22:10 EDT 2021. Contains 346300 sequences. (Running on oeis4.)