Linked Questions

4
votes
3answers
206 views

What are some books that are in the spirit of David A. Cox' “Primes of the Form $x^2+ny^2$”

David A. Cox "Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory, and Complex Multiplication." has a very good (at least to me, and many) methodology. He starts from page 1 asking a simple ...
6
votes
2answers
346 views

Classification of the positive integers not being the sum of four non-zero squares

It is well known that every positive integer is the sum of at most four perfect squares (including $1$). But which positive integers are not the sum of four non-zero perfect squares ($1$ is still ...
16
votes
3answers
294 views

Is there a simple proof that if $(b-a)(b+a) = ab - 1$, then $a, b$ must be Fibonacci numbers? [duplicate]

Consider the identity $(b-a)(b+a) = ab - 1$, where $a, b$ are nonnegative integers. We can also express this identity as $a^2 + ab - b^2 = 1$. This identity is clearly true when $a = F_{2i-1}$ and $...
5
votes
1answer
335 views

Modified Pell equation: $x^2-D y^2 = m$, $m\neq1$.

How does one solve the Diophantine equation $$ x^2-Dy^2=m, $$ where $m$ is some fixed arbitrary integer? I understand that given the fundamental solution to $r^2-D s^2=1$, and any solution to the ...
4
votes
3answers
509 views

Algebraic proof of non-trivial solution to the Pell's equation

Let $d$ be a square-free positive integer, and consider the pairs $(x, y) \in \mathbb{Z}^2$ that satisfy: $$x^2 - dy^2 = 1$$ The existence of a non-trivial solution to this equation (i.e. distinct ...
1
vote
1answer
187 views

Can I check whether integral solutions exist if I know a rational solution?

The pell-like equation $$x^2-101y^2=-71$$ has the rational solution $(x,y)=(\frac{25}{2},\frac{3}{2})$ Can I use this rational point to find out , whether an integral solution exists ? If yes, can ...
3
votes
2answers
105 views

Completeness proofs for the solutions of Diophantine Equations

In general, what are the strategies for showing the completeness of a solution set for Diophantine Equations? For example, take the $\textit{Pell-type}$ equation $x^2 - dy^2 = a$. Say, you have a set ...
6
votes
2answers
321 views

Maps of primitive vectors and Conway's river, has anyone built this in SAGE?

I am attempting to teach number theory from John Stillwell's Elements of Number Theory in the upcoming semester. There are two sections (5.7 and 5.8) which describe the diagrammatic method for the ...
3
votes
1answer
1k views

how to solve binary form $ax^2+bxy+cy^2=m$, for integer and rational $ (x,y)$

solve $ 3x^2+3xy-5y^2=55$ using number theory tools ,i have found the following $\Delta=3^2+4(5)(3)=9+60=69$ $d=69,u=1$ $w_{69}=\frac{1+\sqrt{69}}{2}$ $O_{69}=\theta_{-11}=[1,\frac{1+\sqrt{69}...
2
votes
2answers
177 views

How to prove that the roots of this equation are integers?

Let there be an equation $a^2 + 4ab + b^2 - 121 = 0$ where I want to prove that a,b are integers. Then I want to find whether there are integer values of $b$ for which $a$ is also an integer. Let us ...
0
votes
2answers
391 views

Solutions to Diophantine Equations

I am looking for integer solutions to the equation $$x^2 = 5y^2 + 14y + 1$$ I know that Pell's Equation is of the form $x^2 - ny^2=1$ and that there exist algorithms to solve this equation. I was ...
3
votes
1answer
437 views

General method for determining if $Ax^2 + Bx + C$ is square

Is there a general method for solving Diophantine equations in the form $Ax^2 + Bx + C = k^2$, preferably turning them into Pell's equations, when possible? For example, $2x^2 + x + 1 = k^2$ or $5x^2 +...
2
votes
2answers
855 views

Solving the equation $ x^2-7y^2=-3 $ over integers

I'd like to solve the following Pell equation: $$ x^2-7y^2=-3 $$ Where $x$ and $y$ are integers. I applied the usual procedure, which avoids continued fractions: The two minimal positive integer ...
3
votes
3answers
1k views

The quadratic diophantine $ k^2 - 1 = 5(m^2 - 1)$

Here's the problem. Find the solutions of the following equation: $$ k^2 - 1 = 5(m^2 - 1).$$ Here's my idea: The original equation can be written as: $$ k^2 = 5m^2 - 4 \Longleftrightarrow k^2 - ...
2
votes
3answers
400 views

Linear recurrence solution to Diophantine equation

I have a Diophantine equation of the form: $$ax^2 + bx + c = y^2, \quad x, y \in \mathbb{Z^+}$$ Is it true that there will always be a linear recurrence formula that generates all the solutions for $x$...

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