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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3
votes
1answer
294 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?
1
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0answers
34 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 $|x+y|$? ...
2
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4answers
114 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 ...
2
votes
2answers
86 views

Showing that Pell's equation has at least one integer solution

I'm studying Pell's equations of the form $x^2-dy^2=1$ for $d$ a square-free natural number and $x,y$ integers. In particular, I want to show that such an equation always has a solution. I see that ...
5
votes
4answers
149 views

A caboodle of Pell's equation in one? $x^2+y^2-5xy+5=0$

I saw this twitter post that reads: Find all the pairs of positive integers $(x,y)$ satisfying $$ x^2 + y^2 - 5xy + 5 = 0 . $$ I don't know how to tackle this and I ended up summoning WolframAlpha ...
0
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2answers
137 views

Find all the integer solutions of $x^2-4y^2=1$

Find all the integer solutions of the equation: $x^2-4y^2=1$. I know I can't solve it like a PELL equation because d is a square in this case. Would appreciate your help:)
19
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2answers
565 views

Primes in solutions to Pell-type equations

What is known about primes in solutions to Pell-type equations? In particular, consider the negative Pell equation $x^2 - 5 y^2 = -1$. As far as I've been able to check (in the first $4000$ solutions) ...
2
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2answers
327 views

An infinite family of generalized Pell equations

Let $m,n,x,y,z$ be positive integers such that $y=m(z-1)+1$ and $yz-1=n(y^2-x^2)$. If you fix $m$ and $n$ then this is a generalized Pell equation in $x$ and $z$ (i.e., a quadratic Diophantine ...
5
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0answers
114 views

Difficult implications of a detail in an(other) approach to the simultaneous Pell equations $24a^2+1=t^2$ and $48a^2+1=u^2$

In reviewing & improving my attempt to solve the question on simultaneous Pell-equations (see here in MSE) I came across a detail which has surely wider implications and which I cannot encompass (...
3
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2answers
119 views

Solving $\tau^4 +1=u^2 +v^2$ over the integers

I'm trying to show, that the solutions $t_i$ from Pell's equation $t^2-2s^2=\pm 1$ are not squares (only for the trivial cases). For the even solutions $t_{2i}$ (solutions for the + sign) this is easy,...
0
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1answer
64 views

How to generate infinitely many solutions to $x^2-dy^2 = 4$?

I was looking at how to generate infinitely many solutions to the Pell's equation $x^2-dy^2 = 1$, where $d$ is a square-free positive integer. On https://en.wikipedia.org/wiki/Pell%27s_equation#...
3
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4answers
182 views

Are there Pell Equations $x^2 - dy^2 = n$ that are easy to solve?

Consider the Pell equation $x^2 - dy^2 = n$ where $d$ is a positive non-square integer. Are there examples of special $d$ that makes it easy to solve (obtain non-trivial solutions) the equation for ...
1
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1answer
70 views

Is every integer $z$ representable in Pell form as $x^2 \pm dy^2 =z$?

We know that there are integers that cannot be represented as the sum of two squares (Fermat's Four Square Theorem). We also know that every natural number can be represented as the sum of four ...
0
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1answer
73 views

Method for solving Diophantine equation $ax^2 + bx + c = y^2$

How do I solve the Diophantine equation $ax^2 + bx + c = y^2$? The approach I have so far is to use the transformation $X = 2ax + b$ and $Y = 2y$. Applying this, we get, $X^2 - dY^2 = n$, where $n = b^...
0
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4answers
57 views

Square Triangle numbers and Pell's equation

So i have just started to study number theory and i was asked this question. Now i tried to search online and i found out pell equation can used to solve this question. Now in an online video i saw ...
2
votes
1answer
107 views

Infinitely many solutions of the equation $\frac{x+1}{y}+\frac{y+1}{x} = 4$ [closed]

Prove that there exists infinitely many positive integer solutions in $(x,y)$ to the equation : $$\frac{x+1}{y} + \frac{y+1}{x} = 4$$
0
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2answers
71 views

Recurrence relation solution to negative Pell's equation

I'm trying to determine how to get the sequence of possible solutions for a negative Pell's equation: $$ x^2 - 2y^2=-1 $$ I know that the fundamental solution is $x_1=1$ and $y_1=1$, but I don't know ...
1
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1answer
36 views

The Integral Values For Which The Expression Is A Perfect Square

Given that x is an integer, when does the expression $4x^2 + 80$ form a perfect square? I tried putting $x=4$, got a perfect square but i am not able to calculate how many such cases can be there? ...
1
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2answers
64 views

Question about Pell's equation.

Question. Suppose that $x,y,x',y'$ are positive integers satisfying $x^2-dy^2=\pm 1$ and $(x')^2-d(y')^2=\pm 1$ respectively. Assuming $x<x'$, prove that $y<y'$. Not too sure where to begin. ...
1
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1answer
114 views

A problem involving the largest prime divisor of polynomials of the form $a^2+1$

Question: Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that there exist infinitely many positive integer tuples $a, b, c$ ( all three distinct) such that $P(a)=P(b)=P(c)$. I ...
3
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2answers
183 views

How do we solve pell-like equations?

I need to find all solutions $(x,y)∈Z^2$ to the Pell-like equation $x^2-21y^2= 4$ Method I used to solve above problem:- I solved the pell-equation $x^2-21y^2= 1$ and calculated the possible solutions ...
1
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1answer
75 views

If $(a,b)\in\mathbb N^2$ is a solution of $(x^2-dy^2)^2=1$ with minimal $x$ then $a^2-db^2=-1$.

I need help with the following question: If $d\in\mathbb N$ such as the equation $x^2-dy^2=-1$ has an integer solution. If $(a,b)\in\mathbb N^2$ is a solution of $(x^2-dy^2)^2=1$ with minimal $x$ (...
4
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4answers
109 views

What is the smallest integer $n>1$ for which the mean of the square numbers $1^2,2^2 \dots,n^2 $ is a perfect square?

What is the smallest integer $n>1$ for which the mean of the square numbers $1^2,2^2 \dots,n^2 $ is a perfect square? Initially, this seemed like one could work it out with $AM-GM$, but it doesn't ...
0
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2answers
71 views

If $4 | x^2+y^2+z^2$ then $x, y, z$ are all even

I need to show that if $4 | x^2+y^2+z^2$ then $x, y, z$ are all even. I have absolutely no idea how to solve this question. The second question I have is: If $k,s \in \mathbb N$ then the equation $x^...
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2answers
52 views

$x^n-dy^n$ irreducible?

I am dealing right now with a generalized form of the Pell Equation. In order to use Thue's Theorem I need to know that $x^n-dy^n$ ist irreducible over $\mathbb{Q}$. Somehow I don't get why. I tried ...
0
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2answers
94 views

Can anyone solve this Pell equation?

I have solved the Pell equation $ p^2 - 95 q^2 =1$ . By looking at the convergents corresponding to the simple continued fraction of $\sqrt{95}$ I was able to find the fundamental solution $p=39$ ...
3
votes
2answers
126 views

Pell's equation $x^2-dy^2=4$ always has solutions

I know that Pell's equation $x^2-dy^2=1$ always has solutions and I want using that fact show that $x^2-dy^2=4$ also always has solutions. $$ x^2-dy^2=4\tag{*} $$ I try something like this... Let $(...
0
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1answer
47 views

Database of solutions to this generalised Pell equation.

Does there exist a database of primary solutions to generalised Pell's equations of the form: $$x^2 - 2w^2 = -N$$ for every constant $N \in \mathbb{Z}$?
1
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2answers
87 views

Solvability of the Pell-like equation $x^{2}-dy^{2} = k$

I found some resources that talk about algorithms for finding fundamental solutions to the Pell-like equation $$ x^{2} - dy^{2} = k $$ for $d \in \mathbb{N}$ and $k \in \mathbb{Z}$. However, I'm ...
3
votes
4answers
181 views

When The Sum Of Squares Of Two Consecutive Integers Is Again A Perfect Square?

Find all positive integers $n < 200$, such that $n^2 + (n + 1)^2$ is a perfect square. Well setting this equal to $k^2$ is important. But before that, since all squares $\equiv 0$ or $1$ (mod $3$ $,...
1
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1answer
42 views

How to describe the solutions of a Pell equation which contains a rational number

Let $N(x,y) = x^{2}-dy^{2} $ with d a strictly positive integer and not a square and m an non-zero integer. $$ N(x,y) = m $$ is the general form of Pell's equations which is mostly studied in the ...
1
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2answers
62 views

How many integers solutions does $x^2 + xy - y^2 = 1$ have?

Also, if we know if a binary form represents an integer $n$, is there an algorithm to find all the solutions?
3
votes
2answers
90 views

Integer solutions for $\frac{n(n-1)}2=m^2-1$

I want to find all integer solutions for $$\frac{n(n-1)}2=m^2-1.$$ The only ones I found are $m=\{2,4,64\}$ and $n=\{3,6,91\}$ meaning $m^2-1=\{3,15,4095\}$, but are they the only ones? If not, are ...
0
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1answer
97 views

How to find every primitive solutions of a pell's equation ? And what if one solution is in a specific form?

For example, solve this : $x^2 +2x +3 = 10y^2+11y+12$ with x,y integers I can reduce this equation into this : $Y^2-40X^2=-279$ with $X=x+1$ and $Y=20y+11$ Now I need to find the primitive solutions ...
1
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1answer
156 views

IMO 1988 Q6 $a_n = …$

My question is in bold below. Here is a summary of my working on this problem which is question 6 from the IMO 1988 paper: For $\frac{a^2+b^2}{ab+1}=n$, where $a,b,n \in Z^+$ let $n=x^2$ then for ...
2
votes
3answers
216 views

Prove that the Pell's Equation $x^2 −Dy^2 = 1$ always has a solution where $y$ is a multiple of $41$

$D$ is a positive integer that is not a perfect square Recently I am taking a introductory number theory course and I met this question right after we learned Pell's equation and Diophantine ...
2
votes
2answers
104 views

Integer solutions to the equation $7x^2 = y^2+y+1$

While investigating the related equation $7^n = m^2 + m+1,$ I managed to quite quickly handle the case that $n$ is even. If $n$ is odd, we may let $x = (n-1)/2.$ This now reduces to the question in ...
1
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0answers
47 views

Pell's equation easy bound for the fundamental solution

Let $d>1$ be a non-square integer and consider $x^2 - dy^2 = 1$, with minimal solution $(x_0, y_0)$. Can anyone give an explicit bound for $x_0$ (or $y_0$ or $x_0 + y_0\sqrt{d})$ in terms of $d$, ...
4
votes
3answers
181 views

For $x^2-3y^2=1$ over integers more than $1$, can $\frac{y+1}2$ be square number?

For $x^2-3y^2=1$ over integers more than 1, can $\frac{y+1}2$ be square number? I know that $x^2-3y^2=1$ is one of pell's equation, so I know its general solution. But I know nothing about its ...
0
votes
1answer
111 views

Complicated unusual Pell equation

I am trying to find the five smallest pairs of positive integers p,q that satisfy the unusual Pell equation $$p^2− \frac{251934169}{4} q^2 = 1.$$ One obvious trivial solution is p=1 and q=0 , but ...
1
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1answer
55 views

How do I Transform a Quadratic expression into a Pell Equation?

I have been told that a simple linear transformation (or a change of variables) can transform the quadratic $$x^2+45xy-216y^2$$ into the Pell equation $$p^2−321q^2=1$$ However I have been unable to ...
1
vote
1answer
124 views

Negative Pell equation $x^2-10y^2=-1$: problem with consecutive solutions

Consider the negative Pell equation: $x^2-10y^2=-1$ Its integer positive fundamental solution is $(x_1, y_1)=(3,1)$. On the online solver (https://www.alpertron.com.ar/QUAD.HTM), we know that: $x_{n+...
-2
votes
2answers
71 views

How to solve this Diophantine equation without solving each equation independently?

Find all integral solutions to the equation $x^2 + 4xy - y^2 = m$ with $-5 \leq m \leq 10$. I know that I can set $m = -5$ to $m = 10$ and solve all of the equations independently. But is there any ...
2
votes
1answer
42 views

Given that the point (a,b) lies on the curve $x^2 - 3y^2=1$ find positive integers P,Q,R and S t (Pa + Qb, Ra + Sb) lies on the curve

I am trying to solve this problem. However, I noticed that there are four unknowns while there are only 3 equations that I can use. As a result, guessing might involve. I searched such equations on ...
4
votes
1answer
165 views

Prime solutions to $x^2-2y^2=\pm 1$

It is easy to see that if $(x_0,y_0)$ is a solution to $x^2-2y^2=\pm 1$, then so is $(x_i,y_i)=(3x_{i-1}+4y_{i-1},2x_{i-1}+3y_{i-1})$ (and with the same sign). A comment to this answer points out that ...
2
votes
4answers
114 views

Can anyone help me solve this Pell equation?

I am trying to find the five smallest pairs of positive integers $p,q$ that satisfy the Pell's equation $p^2-321q^2 = 1$. One obvious trivial solution is $p=1$ and $q=0$ , but this does not count. I ...
2
votes
1answer
79 views

Why continued fraction gives initial solution of Pell's equation

According to wikipedia, following algorithm find the smallest solution of $x^2-dy^2=1$. How the validity of this algorithm is shown? Let $α_0 :=\sqrt{d}$. Let $q_i:= ⌊α_i⌋$, $α_{i+1} := 1/(αi − qi)$ ...
0
votes
2answers
265 views

All solutions of a Pell's equation $x^2-2y^2=-1$ [duplicate]

I want to prove the algorithm below produces all solutions of $x^2-2y^2=-1$. Although this algorithms have been already posted to another posts on mathstack exchange, they don't explain why it find ...
1
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0answers
70 views

Minimal solution for some generalized Pell equation

Consider a Pell equation in the form $$X^2-D Y^2=1$$ such that $(a,b)$ is the minimal solution. Now consider the generalized Pell equation $$X^2-DY^2=N^2$$ for some integer $N$: obviously it admits a ...
0
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0answers
55 views

Continued Fractions and Pell's Equation's Smallest Solution

So beginning with the continued fraction $\quad \sqrt{D} = [ \;a_0;\overline { \,a_1, \,\cdots, \phantom {!} a_n , } \cdots \;] \quad$ with the $a_1, \cdots,a_n,$ being the repeating portion. ...

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