Abstract: Let p be a prime number such that p ≡ 1(mod 4), say
p = 1+4k for a positive integer k. Let P = 2k + 1 and Q = k2.
In this paper, we consider the integer solutions of the Pell equation
x2-Py2 = Q over Z and also over finite fields Fp. Also we deduce
some relations on the integer solutions (xn, yn) of it.
Abstract: In this work, we consider the number of integer solutions
of Diophantine equation D : y2 - 2yx - 3 = 0 over Z and
also over finite fields Fp for primes p ≥ 5. Later we determine the
number of rational points on curves Ep : y2 = Pp(x) = yp
1 + yp
2
over Fp, where y1 and y2 are the roots of D. Also we give a formula
for the sum of x- and y-coordinates of all rational points (x, y) on
Ep over Fp.
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.