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

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SUMMARY

The discussion focuses on determining the number of points (x, y) in the field Zp, where p is a prime number, that satisfy the equation y² = x³ + x². The participants explore specific cases, starting with p=2 and p=3, to identify valid points. For p=2, the valid points are (0,0) and (1,0), while for p=3, the analysis involves checking all combinations of (x, y) in Z3. This methodical approach lays the groundwork for understanding the distribution of points on the curve.

PREREQUISITES
  • Understanding of finite fields, specifically Zp.
  • Familiarity with elliptic curves and their properties.
  • Basic knowledge of modular arithmetic.
  • Experience with polynomial equations and their solutions in finite fields.
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  • Explore the properties of elliptic curves over finite fields.
  • Learn about the Hasse-Weil theorem and its implications for counting points on curves.
  • Investigate computational methods for finding points on elliptic curves, such as the Schoof's algorithm.
  • Study examples of other polynomial equations in Zp to compare point distributions.
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Mathematicians, cryptographers, and computer scientists interested in number theory, particularly those working with elliptic curves and finite fields.

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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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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?
 

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