The Number of Rational Points on Elliptic Curves y2 = x3 + a3 on Finite Fields

In this work, we consider the rational points on elliptic curves over finite fields Fp. We give results concerning the number of points Np,a on the elliptic curve y2 ≡ x3 +a3(mod p) according to whether a and x are quadratic residues or non-residues. We use two lemmas to prove the main results first of which gives the list of primes for which -1 is a quadratic residue, and the second is a result from [1]. We get the results in the case where p is a prime congruent to 5 modulo 6, while when p is a prime congruent to 1 modulo 6, there seems to be no regularity for Np,a.

Authors:

• Musa Demirci
• Nazlı Yıldız İkikardeş
• Gökhan Soydan
• İsmail Naci Cangül

Keywords:

• Elliptic curves over finite fields
• rational points
• quadratic residue.

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