Questions tagged [pell-type-equations]

Questions on finding integer solutions to bivariate equations of the form $x^2-Dy^2=a$.

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To get all the (b,a,a+1) Pythagorean triplets [closed]

In $(b,a,a+1)$ I found that $a+a+1=b^2$, in which $a>b$ You guys probably knew this but I randomly thought this thing in $5,12,13$ and then found it in all with $b,a,a+1$ So anyways That means $2a+...
VIBHU's user avatar
  • 73
2 votes
1 answer
78 views

Solvability of this General Pell's Equation

If $p\equiv -1 \pmod{12}$ is a prime, does this general Pell's equation always have integer solutions for $x$ and $y$? $$x^2-3y^2=-p$$ I have tested that this is true for all values of $p$ up to $8\...
PunnyBunny's user avatar
2 votes
3 answers
117 views

How do I find another generalized Pell equation solution after finding the fundamental solution?

If I have a solution $(x_0, y_0)$ to $x^2 - Dy^2 = k$ where D > 1 is not a square, how do I find another solution $(x_1,y_1)$ to $x^2 - Dy^2 = k$ without knowing the solution $(u,v)$ to $u^2 - Dv^...
ender's user avatar
  • 21
6 votes
1 answer
208 views

System of Pell's equations

Let $a$, $b$ and $k$ be positive integers. I want to prove that the only solution of the system $$\left\{\begin{array}{rclr} 2k^2+1&=&a^2&(1)\\ 6k^2+1&=&b^2&(2) \end{array} \...
uvdose's user avatar
  • 107
2 votes
1 answer
144 views

Missing pattern in solvable negative Pell equation

Considering the negative Pell equation $ x^2 - Dy^2 = -1 $, I know that a necessary condition for solvability is that $ D = a^2 + b^2$, with $ a,b $ positive integers. If I fix $ b = 1 $, NPE is ...
user967210's user avatar
0 votes
0 answers
103 views

Analytic Number Theory - distribution of $x^2$ vs. the distribution of $x^2 - 2y^2$

My question originates from Rational Points on Elliptic Curves, (Silverman & Tate), though has little to do with elliptic curves. In chapter $V$: Integer Points on Cubic Curves, section $3$ it ...
Clyde Kertzer's user avatar
1 vote
2 answers
91 views

Command for negative pell equations in PARI-GP or SAGE [closed]

Is there any common command for testing negative Pell equations in PARI-GP or Sage? I searched for this question online and realized that Testing negative Pell equations online is checking solubility, ...
Elei's user avatar
  • 79
2 votes
1 answer
110 views

Is it possible to further generalise Brahmagupta's identity?

Is it possible to generalise Brahmagupta's identity further, by which I mean, for different n, for example take the equations: $$1357 = 37^2 - 3\times2^2$$ $$1357 = 38^2 - 87\times1^2$$ $$1357^2 = ...
Joebloggs's user avatar
  • 170
-1 votes
2 answers
154 views

Solve Diophantine equation $a^2+5ab+3b^2-c^2=0$

Solve Diophantine equation $a^2+5ab+3b^2-c^2=0$ My thoughts are to express it as $(pa+qb)^2 = c^2 $and then solve it as Pell's equation. One solution is $(1,9,17)$.I don't know whether it is a ...
sibillalazzerini's user avatar
0 votes
1 answer
43 views

How to find the general term for Pell's sequence with Euler's method

Hello i try to find the general term for the Pell's sequence with this method called Euler's method . $$ a_{n+1}=2a_{n}+a_{n-1} $$ What I tried here is : $$G(x)=\sum_{n=0}^{\infty }a_{n}x^{n}$$ And i ...
Alexandru Harai's user avatar
1 vote
1 answer
95 views

Integral solutions to $A^2=(u^2+uv+2)/2$, $B^2=(v^2+uv+2)/2$, $C^2=1+uv$

The system of equations $A^2=(u^2+uv+2)/2$, $B^2=(v^2+uv+2)/2$, $C^2=1+uv$ has $A,B,C,u,v$ all integral. Numerical evidence found by brute force computation implies that $uv(u^2-14uv+v^2-16)=0$. ...
coolbear's user avatar
4 votes
2 answers
123 views

Given $d$, how many values of $n$ should I test to get a square of form ${2n^2+d}$

Given $d$, how many values of $n$ should I test to get a square of the form ${2n^2+d}$ Both $d$ and $n$ are a positive integers. There must also be some periodicity in $n$ to jump from the first ...
user avatar
2 votes
1 answer
137 views

For solving a Pell's equation How many iterations should I run before confirming no solution exists.

I am running this piece of code from rosettacode ...
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0 votes
0 answers
55 views

Solvability of $ax^2 +b = y^2$ under integer, and has infinite solutions.

For example, $3x^2 + 4 = y^2$ has infinite integer solutions while $5x^2 + 12 = y^2$ doesn't. At first I thought $b$ has to be the perfect square but $3x^2 + 6 = y^2$ has infinite solutions. So the ...
SGKw's user avatar
  • 181
1 vote
0 answers
46 views

Finding a representation $N = x^2 + ny^2$ efficiently

I want to find a representation of an integer $N$ of the form $N = x^2 + ny^2$. Any single non-trivial representation (i.e. $x \gt 0, y \gt 0, n \ge 1$) suffices. A naive algorithm that I have ...
vvg's user avatar
  • 3,205
1 vote
0 answers
89 views

System of two Pell's equations that might have infinitely many solutions

Does there exist infinitely many integers $(a,b,c)$ verifying simultaneously the two following equations: $$a^2-2b^2=-1$$ and $$a^2-24c^2=1$$ Well I tried this problem with Pell's equation approach $a=...
user avatar
4 votes
2 answers
124 views

Finding a solution to the generalised Pell's equation $x^2-31y^2=5$

I'm trying to find a solution to the generalised Pell's equation $x^2-31y^2=5$. So far I have found the fundamental solution $x=1520,y=273$ to the equation $x^2-31y^2=1$, obtained from calculating the ...
perfect4th's user avatar
2 votes
2 answers
190 views

Integer Solutions to the equation $x^6+5y^2=z^2$

This problem is on Page 194 of “Number Theory and Its History”, by Oystein Ore. I can find infinitely many solutions by letting $x=1$ so that $z^2-5y^2=1$ which has infinitely many solutions. So my ...
wayne thompson's user avatar
2 votes
1 answer
97 views

What's the inverse element in the group of solutions to Pell's equation?

I'm working on a problem from a past exam paper, Define the group operation on the set of solutions of Pell's equation, $$G_d=\{(x,y)\in\mathbb{Z}^2:x^2-dy^2=1\}$$ and show that the group axioms are ...
mjc's user avatar
  • 2,051
3 votes
2 answers
205 views

Is there a short proof that the set of solutions to Pell's equation form an abelian group?

I'm working on a question from a past exam paper, Show that the set of solutions $G_{d,p}$ to Pell's equation $x^2-dy^2=1$ modulo $p$ is a finite abelian group, and compute the order of this group ...
mjc's user avatar
  • 2,051
10 votes
0 answers
255 views

Positive integer solutions of $ab+1=x^2, ac+1=y^2, bc+1=z^2, x+z=2y$

A question essentially the same a this one was asked in MSE 4479792 without any background details and was deleted by the post author after I posted a minimal one sentence answer mentioning five OEIS ...
Somos's user avatar
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1 vote
0 answers
89 views

fundamental solution of Pell’s equation under special conditions

I make the following conjecture from an answer. conjecture Suppose $N$ could be expressed as $$ N= p^2\pm q,\ \ \ \ (p,q \in \mathbb{N},\ 1 \lt q \lt p,\ q\mid 2p) $$ In this case, the fundamental ...
isato's user avatar
  • 426
0 votes
1 answer
108 views

Solution to Pell's equation $a^2 - Nb^2 = 1$ when $a$ is an even integer

Given an integer $a$, show that there is a square-free integer $ N > 1$ and an integer $b$ such that $a^2 - Nb^2 = 1$. I'm not too sure how to go about proving this. I believe I'm supposed to use ...
Patron81's user avatar
-1 votes
1 answer
310 views

How is the Pell-like equation $ax^2-by^2 = c$ solved?

Suppose $a, b$ and $ab$ are non-square and solution exist for the above equation. One way we can solve it is by multiplying $a$ to the equation to get the Pell equation \begin{equation} (ax)^2-aby^2 = ...
Atratrana Suna's user avatar
1 vote
0 answers
103 views

Pell's equation fundamental solution vs. an earlier convergent of $\sqrt{d}$

Let $(p,q)$ be the fundamental solution to Pell's equation $x^2-dy^2=1$, which makes $\frac{p}{q}$ a convergent of $\sqrt{d}$, per the theorem underlying the continued fraction algorithm for solving ...
Szczepan Hołyszewski's user avatar
0 votes
1 answer
80 views

Is there a solution (x,y) to a Pell equation such that $x\equiv \pm 1 \; \mathrm{mod} \; y$?

I have the following question. Let $x^2-Ny^2=1$ be a Pell equation (where $N$ is not a square). Is it possible to find a solution $(x,y)$ such that $x\equiv \pm 1 \; \mathrm{mod} \; y$? If there ...
Fradns's user avatar
  • 460
1 vote
2 answers
241 views

Perfect numbers and Pell's equation

(Disclaimer: The following is a naive attempt to apply the theory of Pell's equations to perfect numbers. Please bear in mind that this is my first time to try solving such an equation.) Let $p^k$ be ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
2 answers
141 views

Generalized Pell's equation where $N$ is perfect square

Are all solutions of the equation $x^2-4My^2=K^2$, multiples of $K$? I am considering $M$ not perfect square. Any tests in Python show be true, but... My code: ...
vandev's user avatar
  • 11
0 votes
1 answer
71 views

Binary quadratic function, when it is equal to a square number?

$x^2+5xy+3y^2=T$ and $\gcd(x,y)=1$ T is a square number. I need to find which x and y values make the function a square number. Gcd is a greatest common divisor function. I converted the function Into ...
eagleofnorth's user avatar
4 votes
2 answers
170 views

Notice that $\sqrt{51}\approx 7+\frac{\sqrt{2}}{10}$

I was doing some other stuff and noticed that: $$\sqrt{51} - \left(7 + \dfrac{\sqrt{2}}{10}\right) = 0.0000070723\,\, (*)$$ and this immediately made me think of their respective continued fractions, ...
dezdichado's user avatar
  • 13.7k
1 vote
1 answer
108 views

Problem (perhaps trivial) with the continued-fraction approach to (generalized) Pell-solutions

I've seen some Q&A's here which put light on the continued-fractions approach to finding solutions for the Pell-equation. I'm trying to get familiar with this method to apply it to my style of ...
Gottfried Helms's user avatar
1 vote
1 answer
134 views

Solutions to the Pell equation $(2x+y)^2-5y^2=4$

The Pell equation $(2x+y)^2-5y^2=4$ has fundamental solution $(1,1)$. I need to find three more solutions, I know tha they are the Fibonacci numbers in ordered pairs $(2,3), (5,8),...$, but I don't ...
ddswsd's user avatar
  • 1,319
7 votes
3 answers
302 views

Do I catch all solutions for the generalized Pell-equation $a^2+b^2 = 2 c^2$ by this matrix-method?

As one question in a somewhat bigger analysis I want to characterize the set of solutions of the generalized Pell-equation in the title: $$a^2+b^2 = 2 c^2 \tag 1$$ I'm not much fluent with the Pell-...
Gottfried Helms's user avatar
2 votes
2 answers
171 views

Fourth Degree Pell Equation

I got stuck at the following of my research problem: Prove that only solution to equation $4b^2-3a^4=1$ for odd positive integers $a$, $b$ is $(1,1)$. I made factorization -->$3a^4 = (2b-1)(2b+1)$...
Atllkks10's user avatar
0 votes
1 answer
60 views

Proving $\frac{L_n(x)^2}{4}-\left(\frac{x^2}{4}+1\right) \cdot F_n(x)^2=(-1)^n$, where $L_n(x)$ and $F_n(x)$ are the Lucas and Fibonacci polynomials

Recently, I found the following identity: $$\frac{L_n(x)^2}{4}-\left(\frac{x^2}{4}+1\right) \cdot F_n(x)^2=(-1)^n $$ where $L_n(x)$ denotes the Lucas polynomials and $F_n(x)$ denotes the Fibonacci ...
Pedja's user avatar
  • 12.8k
1 vote
1 answer
283 views

A general formula for the solutions of the negative Pell's equation

I want to find an explicit formula for the solutions of the negative Pell's equation $x^2 - Dy^2 = -1$ in terms of {a, b} with $D = a^2 + b^2$ I already know that the nth solution is given by the ...
user967210's user avatar
0 votes
0 answers
136 views

When do there exist rational solutions to the generalized Pell equation?

There are many questions on this site asking about integer solutions to the generalized Pell equation $x^2 - dy^2 = n$ for $d$ and $n$ integers and $d$ squarefree. What is known about the existence of ...
user avatar
0 votes
1 answer
166 views

$\lfloor \sqrt{D}\rfloor>q_{l-1}\implies$no odd solution to $x^2-Dy^2=\pm4$?

I try to show that $a_0=\lfloor \sqrt{D}\rfloor>q_{l-1}$ (or $\frac{q_{l-1}}{2}$ if $q_{l-1}$ is even) implies that there is no odd solution to the Pell equation $$x^2-Dy^2=\pm4$$ $D$ is a positive ...
Collag3n's user avatar
  • 2,546
5 votes
5 answers
192 views

Solve in $\mathbf{N}$ the equation $9x^2+p=y^2$

In these days, I have been trying to solve this problem: Let $p \in \mathbf{N}$ a positive large integer ($> 10^9$). Find all $x, y \in\mathbf{N}$ such that: $$9x^2+p=y^2$$ The first approach ...
Matteo's user avatar
  • 6,551
6 votes
2 answers
213 views

$p$-adic structure of Pell-type equations

As an example, consider an integer solution of $ x ^ 2-3y ^ 2 = 13 $. $ y $ that satisfies this equation is $y_k = \frac {(4+ \sqrt {3}) (2+ \sqrt {3}) ^ k-(4- \sqrt {3}) (2- \sqrt {3}) ^ k} { 2 \sqrt ...
Kazsugi's user avatar
  • 147
4 votes
1 answer
385 views

Seeking examples of proofs [by contradiction?] using Pell equation

I’m looking for proofs which use Pell equations as a critical part of the proof. One example is this paper by Robert Phillips: on page 4, he says If $(p, q)$ is one of the infinite number of ...
Kieren MacMillan's user avatar
12 votes
8 answers
571 views

Solving $X^2-6Y^2=Z^3$ in positive integers

I’m trying to solve the Diophantine equation $$X^2-6Y^2=Z^3 \tag{$\star$}$$ in positive integers $x,y,z$. Brute force calculations confirm the naïve intuition that there are many [read: surely ...
Kieren MacMillan's user avatar
2 votes
1 answer
122 views

When does pell equation not have infinite solution?

I am working on a problem and while trying to make some progress, I stumbled across a sub-question. When does $x^2-2y^2 = n$ NOT have infinite solutions for some positive integer $n$. Now I tried to ...
Prathmesh's user avatar
-3 votes
1 answer
124 views

Given $k$, can we find $n$ such that $d$ is a perfect square

Given an odd positive integer $k$. Define $$d=16(k+1)^3(k+2)(n+1)^2+1$$ where $n$ is alos a positive integer. My question is: Given $k$, can we find $n$ such that $d$ is a perfect square.
Safwane's user avatar
  • 3,776
3 votes
2 answers
88 views

Find all $q \in \mathbb{N}$ such that $\frac{q(q+1)}{12}$ is a Perfect square.

Find all $q \in \mathbb{N}$ such that $\frac{q(q+1)}{12}$ is a Perfect square. Trivially we see that $q=3$ is the first candidate. Now let $$\frac{q(q+1)}{12}=r^2$$ $\implies$ $$q^2+q-12r^2=0$$ By ...
Umesh shankar's user avatar
2 votes
1 answer
78 views

Looking for a book/proof for Pell's equation $x^2-2y^2=-1\ \Leftrightarrow\ x+y\sqrt{2}=(1+\sqrt{2})^k$ with odd k

I got this Lemma to Pell's Equation $$ x^2-2y^2=-1\ \Leftrightarrow\ x+y\sqrt{2}=(1+\sqrt{2})^k\ \text{with k odd}\\ x^2-2y^2=+1\ \Leftrightarrow\ x+y\sqrt{2}=(1+\sqrt{2})^k\ \text{with k even} $$ ...
Adn110's user avatar
  • 21
4 votes
0 answers
115 views

Patterns in convergents of continued fraction of $\sqrt{D}$?

First, to give some background: If $D$ is an integer, then the continued fraction of $\sqrt{D}$ is always periodic. For example, the continued fraction of $\sqrt{7}$ is $[2; \overline{1,1,1,4}]$. Also,...
Emily Boyajian's user avatar
1 vote
1 answer
317 views

Find the $d$ such $(d^2+d)x^2-y^2=d^2-1$ has postive integer $(x,y)$

Find all the positive integers $d$ such that $$(d^2+d)x^2-y^2=d^2-1$$ has a positive integer solution $(x,y)$. maybe use Pell equation some result $$x^2-Dy^2=C$$ to solve it?
math110's user avatar
  • 92.8k
1 vote
0 answers
56 views

Nice upper bound on $max{|x|, |y|}$ in terms of $d$ in Pell's Equation

Given Pell's equation $x^2 - dy^2 = 1$, can we come up with a nice upper bound on $max(|x|,|y|)$ in terms of $d$? Also, if it's somehow more tractable, are we able to find an upper bound on $|x+y|$? ...
Chaotic Good's user avatar
2 votes
4 answers
128 views

General term for series $0,2,12,70, 408...$

Given the curves $y=\sqrt{2x^2+1}$ and $x=\sqrt{2y^2+1}$, the first line (line with smallest y coordinate) which intersects the curves with integer coordinates is $x+y=1.$ This line intersects the ...
Sid's user avatar
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