The Polynomial Solutions of Quadratic Diophantine Equation X 2 − p t Y 2 + 2 K t X + 2 p t L t Y = 0

In this study, we consider the number of polynomial solutions of the Pell equation x 2 − p t y 2 = 2 is formulated for a nonsquare polynomial p t using the polynomial solutions of the Pell equation x 2 − p t y 2 = 1 . Moreover, a recurrence relation on the polynomial solutions of the Pell equation x 2 − p t y 2 = 2 . Then, we consider the number of polynomial solutions of Diophantine equation E : X 2 − p t Y 2 + 2 K t X + 2 p t L t Y = 0 . We also obtain some formulas and recurrence relations on the polynomial solution X n , Y n of E .

A Peer Search on Integer Solutions to Quadratic Diophantine Equation with Three Unknowns (Equation)

The non- homogeneous ternary quadratic diophantine (Equation) is analyzed for its patterns of non-zero distinct integral solutions. Various interesting relations between the solutions and special numbers namely polygonal, Pronic and Gnomonic numbers are exhibited.

A Quadratic Diophantine Equation Involving Generalized Fibonacci Numbers

The sequence of the k-generalized Fibonacci numbers ( F n ( k ) ) n is defined by the recurrence F n ( k ) = ∑ j = 1 k F n − j ( k ) beginning with the k terms 0 , … , 0 , 1 . In this paper, we shall solve the Diophantine equation F n ( k ) = ( F m ( l ) ) 2 + 1 , in positive integers m , n , k and l.

ON THE TERNARY QUADRATIC DIOPHANTINE EQUATION 6z2 = 6x2 – 5y2

The ternary homogeneous quadratic equation given by 6z2 = 6x2 -5y2 representing a cone is analyzed for its non-zero distinct integer solutions. A few interesting relations between the solutions and special polygonal and pyramided numbers are presented. Also, given a solution, formulas for generating a sequence of solutions based on the given solutions are presented.