Abstract: In this work, we study elliptic divisibility sequences
over finite fields. Morgan Ward in [14], [15] gave arithmetic theory
of elliptic divisibility sequences and formulas for elliptic divisibility
sequences with rank two over finite field Fp. We study elliptic
divisibility sequences with rank three, four and five over a finite field
Fp, where p > 3 is a prime and give general terms of these sequences
and then we determine elliptic and singular curves associated with
these sequences.
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 y2 = x3 + kx has and
the number of rational points of on Fp. Consider the circle family
x2 + y2 = r2. 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.
Abstract: Let p be a prime number, Fpbe a finite field and let Qpdenote the set of quadratic residues in Fp. In the first section we givesome notations and preliminaries from elliptic curves. In the secondsection, we consider some properties of rational points on ellipticcurves Ep,b: y2= x3+ b2 over Fp, where b ∈ F*p. Recall that theorder of Ep,bover Fpis p + 1 if p ≡ 5(mod 6). We generalize thisresult to any field Fnp for an integer n≥ 2. Further we obtain someresults concerning the sum Σ[x]Ep,b(Fp) and Σ[y]Ep,b(Fp), thesum of x- and y- coordinates of all points (x, y) on Ep,b, and alsothe the sum Σ(x,0)Ep,b(Fp), the sum of points (x, 0) on Ep,b.
Abstract: In this work we study elliptic divisibility sequences over
finite fields. MorganWard in [11, 12] gave arithmetic theory of elliptic
divisibility sequences. We study elliptic divisibility sequences, equivalence
of these sequences and singular elliptic divisibility sequences
over finite fields Fp, p > 3 is a prime.
Abstract: Let k ≥ 1 and t ≥ 0 be two integers and let d = k2 + k be a positive non-square integer. In this paper, we consider the integer solutions of Pell equation x2 - dy2 = 2t. Further we derive a recurrence relation on the solutions of this equation.