THE NUMBER OF LATTICE POINTS ON A LINE
Keywords:
lattice point, line, slope, y-intercept, integer, greatest common divisor, rational number, irrational number, prime number.DOI:
https://doi.org/10.17654/0973563122011Abstract
We study points and lines in the Euclidean plane. A point is called a lattice point if both of its coordinates are integers. The set of lattice points on a given line is either empty, singleton, or (countably) infinite. Number-theoretic considerations involving the parameters $m$ and $b$ from the slope-intercept equation $y = mx + b$ of a non-vertical line are used to characterize the non-vertical lines with infinitely many lattice points, while related methods lead to sufficient conditions for lines to have no lattice points. Many examples are given to motivate and illustrate results, as well as to demonstrate the sharpness of those results. We also provide accessible references to a few standard facts about greatest common divisors and relatively prime integers. Various portions of this note could be used as enrichment material in courses ranging from the middle school to graduate school.
Received: July 1, 2022
Accepted: August 22, 2022
References
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I. Niven and H. S. Zuckerman, An introduction to the theory of numbers, Third edition, Wiley, New York, 1972.
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