Journal of Integer Sequences, Vol. 18 (2015), Article 15.10.8

Rectangles Of Nonvisible Lattice Points

Shanta Laishram
Stat-Math Unit
Indian Statistical Institute
7 S. J. S. Sansanwal Marg
New Delhi, 110016 India

Florian Luca
School of Mathematics
University of the Witwatersrand
P. O. Box X3
Wits 2050
South Africa


A lattice point (0, 0) ≠ (x, y) ∈ Z2 is called visible (from the origin) if gcd(x, y)=1, and nonvisible otherwise. Given positive integers a, b, define M := M(a, b) and N := N(a, b) to be the positive integers M and N having the least value of max(M, N) with the property that gcd(M-i, N-j) > 1 for all 1 ≤ ia and 1 ≤ jb. We give upper and lower bounds for M, N.

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(Concerned with sequence A052157.)

Received May 7 2015; revised versions received October 12 2015; October 24 2015; October 27 2015. Published in Journal of Integer Sequences, October 29 2015.

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