Number of lattice points between y=ax+b and y=x^2?

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Discussion Overview

The discussion revolves around determining the number of lattice points between the linear equation y=ax+b and the quadratic equation y=x^2. Participants explore whether a closed-form solution exists and the conditions under which the count of lattice points is considered.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions if there is a closed-form solution for the number of lattice points between the two curves.
  • Another participant suggests that the straight line can intersect the parabola at most twice and discusses the interpretation of the area between the two curves, raising the possibility of counting lattice points on the curves themselves or within the area formed.
  • A further clarification is made regarding the integral lattice points specifically within the area formed between the two curves.
  • One participant proposes a method involving solving the quadratic equation derived from setting the two equations equal, followed by summing values to count lattice points, while also noting adjustments to exclude points on the lines themselves.

Areas of Agreement / Disagreement

Participants express differing interpretations of the problem, particularly regarding whether to include lattice points on the curves or only those within the area. There is no consensus on a closed-form solution or the exact counting method.

Contextual Notes

Assumptions about the values of a and b being integers are mentioned, but the implications of these assumptions on the counting process remain unresolved. The discussion does not clarify the definitions of the area or the specific conditions under which lattice points are counted.

SeventhSigma
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Is there a nice closed-form for this?
 
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Since a straight line intercepts a parabola in at most two points, I believe your looking for the number of lattice points on both the curve and straight line that surround the area between the two lines. Is that right? Otherwise the answer would be one of either 0,1,2 or infinity. Infinity would be if you include all lattice points on the parabola, not just those bordering the area. Then again you may be looking for the number of coordinite points within the area so I am unsure what you are looking for. Also, should we assume that a and b are integers?
 
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The integral lattice points in the area formed between the two curves
 
SeventhSigma said:
Is there a nice closed-form for this?

Solve the Quadratic equation x^2 - ax - b = 0 to get x small and x large. Then from the ceiling of x small to the floor of x large, sum ax+b +1 - x^2 for integer values of x. I think that will give you your sum.

Final Edit: the above includes the lattice points of the lines, to exclude them, sum the value "ax + b -1 - x^2" for integer values between x-small and x-large. Both sums assume that "a" and "b" are integers.
 
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