For each positive integer n ≤ 7 we describe all Lucas sequences with (P,Q) = 1 having the property that Un(P,Q) is a perfect square. The arguments are elementary. We also find all Lucas sequences such that U8(P,Q) is a perfect square. This reduces to a number of problems of similar type, namely, finding all points on an elliptic curve defined over a quartic number field subject to a “Q-rational...

For each positive integer n ≤ 7 we describe all Lucas sequences with (P,Q) = 1 having the property that Un(P,Q) is a perfect square. The arguments are elementary. We also find all Lucas sequences such that U8(P,Q) is a perfect square. This reduces to a number of problems of similar type, namely, finding all points on an elliptic curve defined over a quartic number field subject to a “Q-rational...

In this paper we consider the Suzuki curve y + y = x0(x + x) over the field with q = 2 elements. The automorphism group of this curve is known to be the Suzuki group Sz(q) with q(q − 1)(q + 1) elements. We construct AG codes over Fq4 from a Sz(q)-invariant divisor D, giving an explicit basis for the Riemann-Roch space L(lD) for 0 < l ≤ q − 1. These codes then have the full Suzuki group Sz(q) as...

Let E be an elliptic curve defined over Fq, the finite field of q elements. We show that for some constant η > 0 depending only on q, there are infinitely many positive integers n such that the exponent of E(Fqn), the group of Fqn-rational points on E, is at most q exp ( −n log logn ) . This is an analogue of a result of R. Schoof on the exponent of the group E(Fp) of Fp-rational points, when a...

We count the number of isogeny classes of Edwards curves over finite fields, answering a question recently posed by Rezaeian and Shparlinski. We also show that each isogeny class contains a complete Edwards curve, and that an Edwards curve is isogenous to an original Edwards curve over IFq if and only if its group order is divisible by 8 if q ≡ −1 (mod 4), and 16 if q ≡ 1 (mod 4). Furthermore, ...

The problem of the form of the 'arctic' curve of the six-vertex model with domain wall boundary conditions in its disordered regime is addressed. It is well-known that in the scaling limit the model exhibits phase-separation, with regions of order and disorder sharply separated by a smooth curve, called the arctic curve. To find this curve, we study a multiple integral representation for the em...

We study the problem of the existence of arithmetic progressions of three cubes over quadratic number fields Q( √ D), where D is a squarefree integer. For this purpose, we give a characterization in terms of Q( √ D)-rational points on the elliptic curve E : y = x − 27. We compute the torsion subgroup of the Mordell–Weil group of this elliptic curve over Q( √ D) and we give an explicit answer, i...

The pressure-flow (P-Q) curve has been widely used in many studies to describe the effects of various factors on vascular hemodynamics. It is not clear, however, whether unilateral ischemia-reperfusion (IR) alters the P-Q curve of the rat lung. In this study, we developed an in vivo P-Q curve using the unilateral (left) rat lung before and after IR. Animals were divided into two groups: sham an...

Let E be an elliptic curve defined over Q given by an affine Weierstrass equation of the form (1) E : y = x + ax + b (a, b ∈ Z, x, y ∈ Q). Reducing the elliptic curve (1) modulo a sufficiently large prime p, we obtain an elliptic curve Ẽp over Fp. Considering an infinite sequence of elliptic curves Ẽp, we map the point (x, y) of them into the unit square [0, 1) via the mapping (x, y) 7→ ( x p ,...