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: Let k, t, d be arbitrary integers with k ≥ 2, t ≥ 0 and
d = k2 - k. In the first section we give some preliminaries from
Pell equations x2 - dy2 = 1 and x2 - dy2 = N, where N be any
fixed positive integer. In the second section, we consider the integer
solutions of Pell equations x2 - dy2 = 1 and x2 - dy2 = 2t. We
give a method for the solutions of these equations. Further we derive
recurrence relations on the solutions of these equations
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 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.