Linked Questions

17
votes
3answers
5k views

How many elements in a number field of a given norm?

Let $K$ be a number field, with ring of integers $\mathcal{O}_k$. For $x\in \mathcal{O}_K$, let $f(x) = |N_{K/\mathbb{Q}}(x)|$, the (usual) absolute value of the norm of $x$ over $\mathbb{Q}$. ...
15
votes
5answers
1k views

Fibonacci number identity.

How do I see that $f_{n+1}f_{n-1} = f_n^2 + (-1)^n$, $n \ge 2$, where $f_1 = 1$, $f_2 = 1$, and $f_{n+2} = f_{n+1} + f_n$ for $n \in \mathbb{N}$?
36
votes
1answer
2k views

Why does this test for Fibonacci work?

In order to test if a number $A$ is Fibonacci, all we need to do is compute $5A^2 + 4$ and $5A^2 -4$. If either of them is a perfect square, the number is Fibonacci, otherwise not. Why does this test ...
16
votes
3answers
298 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 $...
2
votes
2answers
903 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 ...
6
votes
2answers
360 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 ...
4
votes
3answers
1k views

Solving Pell's equation $x^2-5y^2=\pm4$ using elementary methods.

Solve Pell's equation $x^2-5y^2=\pm4$. This equation arises when I tried to prove that the units of $\mathbb{Z}[\varphi]$, where $\varphi=\frac{1+\sqrt{5}}{2}$ is the golden ratio, are of the form $\...
4
votes
3answers
554 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 ...
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}...
4
votes
3answers
213 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 ...
3
votes
1answer
488 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 +...
5
votes
2answers
175 views

Pell Equations: $a^2+4=5b^2$

This is a challenge problem in the Pell Equations chapter of my number theory book, but I'm not seeing the connection to Pell Equations. The Pell Equation with the coefficient $5$ is $5b^2+1=a^2$, but ...
0
votes
2answers
411 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 ...
2
votes
2answers
195 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 ...
2
votes
1answer
254 views

Fermat's Difference Equation / Pell's Equation

Solve the equation $x^2-5y^2=4$ in positive integers. I know that the base case is $(3,1)$, but I don't know how to get the general form (like how you do it at a pell equation) I have the basic ...

15 30 50 per page