Abstract—In elliptic curve theory, number of rational points on elliptic curves and determination of these points is a fairly important problem. Let p be a prime and Fp be a finite field and k ∈ Fp. It is well known that which points the curve y 2 = x 3 + kx has and the number of rational points of on Fp. Consider the circle family x 2 + y 2 = r 2. It can be interesting to determine common points of these two curve families and to find the number of these common points. In this work we study this problem. Keywords—Elliptic curves over finite fields, rational points on elliptic curves and circles. I
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