# Pell equation

## Top PDF Pell equation: ### ON THE NEGATIVE PELL EQUATION

y , where D is a given positive square-free integer is known as pell equation and is one of the oldest Diophantine equation that has interesting mathematicians all over the world, since antiquity, J.L. Lagrange proved that the positive pell equation y 2  Dx 2  1 has infinitely many distinct integer solutions whereas the negative pell equation ### Observation on the Negative Pell Equation

y have integer solutions. In this context, one may refer [4-10] for a few negative pell equations with integer solutions. In this communication, the negative pell equation given by y 2  12 x 2  23 is considered and analysed for its integer solutions. A few interesting relations among the solutions are given. Employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas and parabolas. ### A Digital Signature Scheme Based on Pell Equation

Abstract:Elliptic curve digital signature algorithm (ECDSA) is well established digital signature scheme based on the discrete logarithms problems. On the other hand, many cryptosystem has been designed using the Pell Equation. We apply the idea of ECDSA on the solution space of Pell equation to design a digital signature scheme. In this paper we compare the security of our signatures scheme to DSA and ECDSA. Our scheme is as secure as conventional DSA. We show that the signatures scheme based on Pell equation is more efficient than its analogue to elliptic curve i.e. ECDSA. ### The Pell Equation X2 Dy2 = ± k2

For completeness we recall that there are many papers in which are considered different types of Pell's equation. Many authors such as Tekcan , Kaplan and Williams , Matthews , Mollin, Poorten and Williams , Stevenhagen  and the others consider eome specific Pell equations and their integer solutions. A. Tekcan in , considered the equation x 2  Dy 2 = 4  , and he ob- tained some formulas for its integer solutions. He mentioned two conjecture which was proved by A. S. Shabani . In this paper we extend the work of A. Tek- can by considering the Pell equation x 2  Dy 2 =  k 2 when D  1 be a positive non-square and k  2 , we obtain some formulas for its integer solutions. ### On The Positive Pell Equation \$y^2=90 x^2+31\$

In this paper, we have presented infinitely many integer solutions for the positive pell equation y 2 = 90 x 2 + 31 .As the binary quadratic Diophantine equations are rich in variety, one may search for the other choices of pell equations and determine their integer solutions along with suitable properties. ### On The Negative Pell Equation

Diophantine equation of the form y 2  Dx 2  1 , where D  0 and square free, is known as negative pell equation. In general, the general form of negative pell equation is represented by y 2  Dx 2  N , N  0 , D  0 and square free. It is known that negative pell equations y 2  3 x 2  1 , y 2  7 x 2  4 have no integer solutions whereas y 2  65 x 2  1 , y 2  202 x 2  1 have integer solutions. It is observed that the negative pell equation do not always have integer solutions. For negative pell equations with integer solutions, one may refer [1-12]. ### On the Positive Pell Equation y^2=8x^2+49

The binary quadratic equations of the form y 2  Dx 2  1 where D is non-square positive integer has been selected by various mathematicians for its non-trivial integer solutions when D takes different integral values[1-4].For an extensive review of various problems, one may refer[5-10]. In this communication, yet another interesting equation given by y 2  8 x 2  49 is considered and infinitely many integer solutions are obtained. A few interesting properties among the solutions are presented. ### ON THE NEGATIVE PELL EQUATION y2=102x2-18

The binary quadratic equations of the form y 2  Dx 2  1 where D is non-square negative integer has been selected by various mathematicians for its non-trivial integer solutions when D takes different integral values[1-4].For an extensive review of various problems, one may refer[5-10]. In this communication, yet another interesting equation given by y 2  102 x 2  18 is considered and infinitely many integer solutions are obtained. A few interesting properties among the solutions are presented. ### ON THE P0SITIVE PELL EQUATION y2=34x2+18

The binary quadratic equations of the form y 2  Dx 2  1 where D is non-square negative integer has been selected by various mathematicians for its non-trivial integer solutions when D takes different integral values[1-4].For an extensive review of various problems, one may refer[5-10]. In this communication, yet another interesting equation given by y 2  34 x 2  18 is considered and infinitely many integer solutions are obtained. A few interesting properties among the solutions are presented. ### On The Negative Pell Equation Y2 = 30x2 - 45

The binary quadratic equations of the form y 2  Dx 2  1 where D is non-square positive integer has been selected by various mathematicians for its non-trivial integer solutions when D takes different integral values[1- 4].For an extensive review of various problems, one may refer[5-10]. In this communication, yet another interesting equation given by y 2  30 x 2  45 is considered and infinitely many integer solutions are obtained. A few interesting properties among the solutions are presented. ### On the Negative Pell Equation y2 = 48x2 23

The binary quadratic Diophantine equations (both homogeneous and non-homogeneous) are rich in variety. In [1-17] the binary quadratic non-homogeneous equations representing hyperbolas respectively are studied for their non-zero integral solutions. This communication concerns with yet another binary quadratic equation given by y 2  48 x 2  23 .The recurrence relations satisfied by the solutions x and y are given. Also a few interesting properties among the solutions are exhibited. ### Construction of Strong Diophantine Quadruples and Pell Equation

It is worth mentioning here that, when k=0, (6) represents Strong diophantine quadruples with property D(1) in integers, whereas when k>0, (6) represents Strong diophantine quadruple[r] ### On the Positive Pell Equation Y2=35x2+29

----------------------------------------------------------------------***--------------------------------------------------------------------- Abstract – The binary quadratic Diophantine equation represented by the positive Pellian y 2  35 x 2  29 is analyzed for its non-zero distinct integer solutions. A few interesting relations among the solutions are given. Employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas and parabolas. ### OBSERVATION ON THE POSITIVE PELL EQUATION

The binary quadratic Diophantine equations of the form ax 2  by 2  N ,  a , b , N  0  are rich in variety and have been analyzed by many Mathematicians for their respective integer solutions for particular values of a, and b N . In this context, one may refer [1-13]. This communication concerns with the problem of obtaining non-zero distinct integer solutions to the binary quadratic equation given by y 2  2 x 2  18 representing hyperbola. A few interesting relations among its solutions are presented. Knowing an integral solution of the given hyperbola, integer solutions for other choices of hyperbolas and parabolas are presented. Also, employing the solutions of the given equation, special Pythagorean triangle is constructed. ### Observation on the Negative Pell Equation

Abstract: The binary quadratic equation represented by the negative pellian y 2  40 x 2  36 is analyzed for its distinct integer solutions. A few interesting relations among the solutions are given. Employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas and parabolas. Also, the relations between the solutions and special figurate numbers are exhibited. ### On the Positive Pell Equation

The binary quadratic equations of the form y 2  Dx 2  1 where D is non-square positive integer has been selected by various mathematicians for its non-trivial integer solutions when D takes different integral values[1-4] .For an extensive review of various problems, one may refer[5-10]. In this communication, yet another interesting equation given by y 2  17 x 2  8 is considered and infinitely many integer solutions are obtained. A few interesting properties among the solutions are presented ### On the Positive Pell Equation

International Journal for Research in Applied Science & Engineering Technology IJRASET ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 6.887 Volume 7 Issue I, Jan 2019- Available at [r] ### On the Positive Pell Equation

A binary quadratic equation of the form y 2  Dx 2  1 where D is non-square positive integer has been studied by various mathematicians for its non-trivial integral solutions when D takes different integral values [1-2]. For an extensive review of various problems, one may refer [3-12]. In this communication, yet another interesting hyperbola given by y 2  32 x 2  41 is considered and infinitely many integer solutions are obtained. A few interesting properties among the solutions are obtained. ### Observation on the Positive Pell Equation

The binary quadratic equation of the form y 2  Dx 2  1 where D is non-square positive integer, has been selected by various mathematicians for its non-trivial integer solutions when D takes different integral values [1-4]. For an extensive review of various problems, one may refer [5-16]. In this communication, yet another an interesting equation given by y 2  15 x 2  10 is considered and infinitely many integer solutions are obtained. A few interesting properties among the solutions are presented. ### On Pell, Pell Lucas, and balancing numbers

said that Euler (1707-1783) mistakenly attributed Brouncker’s (1620-1684) work on this equation to Pell. However, the equation appears in a book by Rahn (1622-1676), which was certainly written with Pell’s help: some say that it is entirely written by Pell. Perhaps Euler knew what he was doing in naming the equation. In 1657, Fermat stated (without giving proof ) that the positive Pell equation x 2 – dy 2 = 1 has an inﬁnite number of solu-