Simultaneous Pell equations x2 - my2 = 1 and y2 - pz2 = 1

Pell equation is a special type of Diophantine equations of the form x2 − my2 = 1, where m is a positive non-square integer. Since m is not a perfect square, then there exist infinitely many integer solutions (x, y) to the Pell equation. This paper will discuss the integral solutions to the simultaneous Pell equations x2 − my2 = 1 and y2 − pz2 = 1, where m is square free integer and p is odd prime. The solutions of these simultaneous equations are of the form of (x, y, z, m) = (yn2t±1, yn, zn, yn2t2±2t) and (y2n/2 t ±1, yn, zn, y2n/4 t2) for yn odd and even respectively, where t ∈ N

Simultaneous Pell equations

This paper will discuss the solutions on the simultaneous Pell equations x2 − my2 = 1 and y2 − 11z2 = 1 where m is square free. By looking at the pattern of the solutions, some theorems will be developed. The solutions to these simultaneous equations are (x, y, z, m) = (50i − 1, 10, 3, mi) and (50i + 1, 10, 3, mi) for some expressions of mi where i is natural number

Complete solutions of the simultaneous Pell's equations (a2+2)x2−y2=2 (a^2+2)x^2-y^2 = 2 and x2−bz2=1 x^2-bz^2 = 1

In this paper, we consider the simultaneous Pell equations $(a^2+2)x^2-y^2 = 2$ and $x^2-bz^2 = 1$ where $a$ is a positive integer and b > 1 is squarefree and has at most three prime divisors. We obtain the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution in positive integers. When a solution exists, assuming the positive solutions of the Pell equation $(a^2+2)x^2-y^2 = 2$ are $x = x_m$ and $y = y_m$ with $m\geq 1$ odd, then the only solution of the system is given by $m = 3$ or $m = 5$ or $m = 7$ or $m = 9$

Several Methods for Solving Simultaneous Fermat-Pell Equations

In our previous papers [12] and [13] , we have exhibited the structure of certain real bicyclic biquadratic fields and as a byproduct solved the simultaneous Fermat-Pell equations x^2-3y^2=1, y^2-2z^2=-1 have only one non-negative integer solution: (x, y, z)=(2, 1, 1). In this paper, we shall investigate similar simultaneous Fermat-Pell equations and solve them by several different methods

Simultaneous pell equations x²-my² = 1 and y²-pz² = 1

Pell equation is a special type of Diophantine equations of the form x²−my²= 1, where m is a positive non-square integer. Since m is not a perfect square, then there exist infinitely many integer solutions(x, y)to the Pell equation. This paper will discuss the integral solutions to the simultaneous Pell equationsx²−my²= 1 and y²−pz²= 1, where m is square free integer and p is odd prime. The solutions of these simultaneous equations are of the form of(x, y, z, m) = (yn²t±1, yn, zn, yn²t²±2t)and(y²n/²t±1, yn, zn, y²n/4t²±t) for yn odd and even respectively, where t ∈ N

When are Multiples of Polygonal Numbers again Polygonal Numbers?

Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation P=mP' is satisfied by infinitely many pairs of triangular numbers P, P'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit in quadratic number fields, finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP', P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P, P', P'' of polygonal numbers, and give examples of such solutions.Comment: 17 pages, 1 figure, 2 tables. New version added a table of solutions to the second proble

On the intersections of Fibonacci, Pell, and Lucas numbers

We describe how to compute the intersection of two Lucas sequences of the forms $\{U_n(P,\pm 1) \}_{n=0}^{\infty}$ or $\{V_n(P,\pm 1) \}_{n=0}^{\infty}$ with $P\in\mathbb{Z}$ that includes sequences of Fibonacci, Pell, Lucas, and Lucas-Pell numbers. We prove that such an intersection is finite except for the case $U_n(1,-1)$ and $U_n(3,1)$ and the case of two $V$-sequences when the product of their discriminants is a perfect square. Moreover, the intersection in these cases also forms a Lucas sequence. Our approach relies on solving homogeneous quadratic Diophantine equations and Thue equations. In particular, we prove that 0, 1, 2, and 5 are the only numbers that are both Fibonacci and Pell, and list similar results for many other pairs of Lucas sequences. We further extend our results to Lucas sequences with arbitrary initial terms