The Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin with radius r. This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area. Carl Friedrich Gauss made the first progress on a solution, giving the problem its name. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementThe problem asks how many pairs of integers m and n lie inside a circle of radius r centered at the origin, that is, how many integer solutions satisfy m squared plus n squared is less than or equal to r squared. That count is closely approximated by the circle's own area, pi times r squared, so the real content of the problem is how tightly the difference between the true count and that area can be bounded as r grows. 1 Progress Toward ResolutionNo unconditional result pins down the true size of the error term between the lattice point count and the circle's area. The known bounds bracket it from both sides: a lower bound established by Hardy and Landau in 1915, and an upper bound proved by Martin Huxley in 2000. The conjectured true exponent for the tightest possible bound remains open. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
Sources
1. Gauss Circle Problem (Wikipedia)
Wikimedia FoundationLead section
the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r
Known bounds paragraph
with the lower bound from Hardy and Landau in 1915, and the upper bound proved by Martin Huxley in 2000.
Bounds on a solution and conjecture section, conjectured bound sentence
It is conjectured that the correct bound is |E(r)| = O(r^(1/2 + ε)).
View the Source Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.