The theory of Pell’s equation has a long history, as can be seen from the huge amount of references collected in Dickson [Dic1920], from the two books on its history by Konen [Kon1901] and Whitford [Whi1912], or from the books by Weber [Web1939], Walfisz [Wal1952], Faisant [Fai1991], and Barbeau [Bar2003]. For the better part of the last few centuries, the continued fractions method was the und...

We obtain some new identities for the generalized Fibonacci polynomial by a approach, namely, Q(x) matrix.
 These including Cassini type identity and Honsberger formula can be applied to polynomial
 sequences such as polynomials, Lucas Pell Pell-Lucas polynomials so on, which
 generalize previous results in references.

In this study, a new sequence called the complex-type k -Pell number is defined. Also, we give properties of such as generating matrix, function, combinatorial representations, exponential representation, sums, permanental and determinantal Binet formula. Then, determine periods recurrence according to modulo id="M3"> <m...

then the Fibonacci numbers are given by Fn = Fn (1), and the Pell numbers are Pn = Fn(2). Cohn [4] has proved that the only perfect squares among the sequences {Fn(a)}9 a odd, are 0 and 1, and whenever a = k, a itself. Certain cases are known for a even [5]. The cited results of Cohn and Ljunggren mean that Conjectures 2.3,3.2, and 4.2 of [3] are true, and that the earlier results can be streng...

Let R and S be positive integers with R < S. We shall call the simultaneous Diophantine equations x −Ry = 1, z − Sy = 1 simultaneous Pell equations in R and S. Each such pair has the trivial solution (1, 0, 1) but some pairs have nontrivial solutions too. For example, if R = 11 and S = 56, then (199, 60, 449) is a solution. Using theorems due to Baker, Davenport, and Waldschmidt, it is possible...

We prove that powers of 3-netted matrices (the entries satisfy a third-order recurrence δai,j = αai−1,j+βai−1,j−1+γai,j−1) preserve this property of nettedness, that is the entries of the e-th power satisfy δea (e) i,j = αea (e) i−1,j +βea (e) i−1,j−1+γea (e) i,j−1, where the coefficients are all instances of the same sequence, xe+1 = (β + δ)xe − (βδ + αγ)xe−1. Also, we find a matrix Tn(m) and ...

Leonhard Euler called (1) Pell’s Equation after the English mathematician John Pell (1611-1685). This terminology has persisted to the present day, despite the fact that it is well known to be mistaken: Pell’s only contribution to the subject was the publication of some partial results of Wallis and Brouncker. In fact the correct names are the usual ones: the problem of solving the equation was...

in this paper‎, ‎we define two $n$-square upper hessenberg matrices one of which corresponds to the adjacency matrix of a directed pseudo graph‎. ‎we investigate relations between permanents and determinants of these upper hessenberg matrices‎, ‎and sum formulas of the well-known pell and jacobsthal sequences‎. ‎finally‎, ‎we present two maple 13 procedures in order to calculate permanents of t...

An arbitrary quadratic diophantine equation with two unknowns can be reduced to a Pell-type equation. How can such equations be solved? Recall that the general solution of a linear diophantine equation is a linear function of some parameters. This does not happen with general quadratic diophantine equations. However, as we will see later, in the case of such equations with two unknowns there st...