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

[SeventhSigma]

Is there a nice closed-form for this?

 

[ramsey2879]

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?

 

[SeventhSigma]

The integral lattice points in the area formed between the two curves

 

[ramsey2879]

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.

 

[blue_raver22]

The number of lattice points between two functions, y=ax+b and y=x^2, can be found by setting the two equations equal to each other and solving for x. This will give us the points where the two functions intersect. However, since these are both continuous functions, there will be an infinite number of points between them.

To find the exact number of lattice points, we would need to define the range of x values and then round the resulting x values to the nearest integer to get the lattice points. This can be done for a specific range of x values, but it would be difficult to find a closed-form solution that works for all possible values of a and b.

Additionally, the number of lattice points will vary depending on the values of a and b. If a is a large number, the two functions will have a small range of intersection, resulting in fewer lattice points. On the other hand, if a is a small number, the two functions will have a larger range of intersection, resulting in more lattice points.

In conclusion, while it is possible to find the number of lattice points between these two functions, there is not a simple closed-form solution that works for all values of a and b. The exact number of lattice points will depend on the specific values of a and b and the chosen range of x values.