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SeventhSigma
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Is there a nice closed-form for this?
SeventhSigma said:Is there a nice closed-form for this?
The formula for finding the number of lattice points between two curves is given by the difference of the number of lattice points on the two curves. This can be calculated by taking the integer value of the highest point on the lower curve and subtracting the integer value of the lowest point on the upper curve.
The slope of the line, or the value of 'a' in the equation y=ax+b, determines the spacing between the lattice points. A steeper slope will result in a larger spacing between points, while a smaller slope will result in a smaller spacing between points.
No, the number of lattice points between two curves will always be a positive integer. This is because lattice points are defined as points with integer coordinates, and the formula for calculating the number of lattice points between two curves only considers the difference between the number of points on the two curves, which will always be a positive value.
The shape of the curves, specifically the concavity and steepness, can greatly impact the number of lattice points between them. A more concave curve will have a larger number of lattice points than a flatter curve, while a steeper curve will have a larger spacing between points than a shallower curve.
There is no specific limit to the number of lattice points between two curves. However, as the distance between the two curves decreases, the number of lattice points between them will increase. In theory, there could be an infinite number of lattice points between two curves that are very close together.