Abstract: Let F(x, y) = ax2 + bxy + cy2 be a positive definite
binary quadratic form with discriminant Δ whose base points lie on
the line x = -1/m for an integer m ≥ 2, let p be a prime number
and let Fp be a finite field. Let EF : y2 = ax3 + bx2 + cx be an
elliptic curve over Fp and let CF : ax3 + bx2 + cx ≡ 0(mod p) be
the cubic congruence corresponding to F. In this work we consider
some properties of positive definite quadratic forms, elliptic curves
and cubic congruences.
Abstract: In this paper, we derive some algebraic identities on
right and left neighbors R(F) and L(F) of an indefinite binary
quadratic form F = F(x, y) = ax2 + bxy + cy2 of discriminant
Δ = b2 -4ac. We prove that the proper cycle of F can be given by
using its consecutive left neighbors. Also we construct a connection
between right and left neighbors of F.
Abstract: Let D = 1 be a positive non-square integer and let δ = √D or 1+√D 2 be a real quadratic irrational with trace t =δ + δ and norm n = δδ. Let γ = P+δ Q be a quadratic irrational for positive integers P and Q. Given a quadratic irrational γ, there exist a quadratic ideal Iγ = [Q, δ + P] and an indefinite quadratic form Fγ(x, y) = Q(x−γy)(x−γy) of discriminant Δ = t
2−4n. In the first section, we give some preliminaries form binary quadratic forms, quadratic irrationals and quadratic ideals. In the second section, we obtain some results on γ, Iγ and Fγ for some specific values of Q and P.