Questions tagged [pell-type-equations]
Questions on finding integer solutions to bivariate equations of the form $x^2-Dy^2=a$.
309
questions
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
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
vote
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
vote
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
vote
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
votes
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
vote
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
votes
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
votes
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^...
-1
votes
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
votes
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
votes
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
vote
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
vote
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
vote
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
votes
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
vote
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
vote
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
vote
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
vote
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
votes
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. ...
