Number of Points on Zp Curve for y^2 = x^3 + x^2

In summary, the problem is to determine the number of points (x, y) with x, y in Zp such that y^2 = x^3 + x^2, where p is a prime number. An approach is to look at simple examples and see how many of the possible points satisfy the equation.
  • #1
Markjdb
31
0
Let p be a prime number. Let Zp denote the field of integers modulo p. Determine the
number of points (x, y) with x, y in Zp such that y^2 = x^3 + x^2.

I just don't really have any idea how to approach this; the last problem was to find all rational points on the above curve, which I did, but I'm not quite sure where to start with this one.
 
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  • #2
An obvious way to start is to look at simple examples. If p= 2, then Zp consists only of 0 and 1 so the only possible points are (0,0), (1, 0), (0,1), and (1,1). How many of those satisfy the equation? If p= 3, Zp= Z3 consists of 0, 1, and 2 so there are 9 possible points. How many of them satisfy the equation?
 

1. How many points are there on the Zp curve for y^2 = x^3 + x^2?

The number of points on the Zp curve for y^2 = x^3 + x^2 depends on the value of p, the prime number used to define the finite field. In general, there can be anywhere from 1 to p+1 points on the curve.

2. How do you calculate the number of points on the Zp curve for y^2 = x^3 + x^2?

To calculate the number of points on the Zp curve for y^2 = x^3 + x^2, one can use the Hasse's theorem. This theorem states that the number of points on the curve is bounded by (p+1)^2 - 4p. However, this is just an upper bound and the actual number of points may be less than this value.

3. Can there be infinitely many points on the Zp curve for y^2 = x^3 + x^2?

No, there cannot be infinitely many points on the Zp curve for y^2 = x^3 + x^2. The curve is defined over a finite field, so the number of points is also finite.

4. How does the number of points on the Zp curve for y^2 = x^3 + x^2 change with different values of p?

The number of points on the Zp curve for y^2 = x^3 + x^2 can vary greatly with different values of p. For example, when p = 2, there is only one point on the curve, but when p = 7, there are 9 points. In general, as p increases, the number of points also increases, but it is not a linear relationship.

5. Are there any patterns or relationships between the number of points on the Zp curve for y^2 = x^3 + x^2 and the value of p?

Yes, there are some patterns and relationships between the number of points on the Zp curve for y^2 = x^3 + x^2 and the value of p. For example, the number of points is always even when p is an odd prime. Additionally, there are certain values of p, known as supersingular primes, that have a special relationship with the number of points on the curve. However, there is not a simple formula or pattern that can determine the number of points for any given p.

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