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
KEYWORD
nonn
AUTHOR
Christopher E. Thompson, Dec 08 2019
STATUS
approved