Intereting Posts

Why $\lim\limits_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=\lim\limits_{t\to 0}\frac{\sin t}{t}$?
Special orthogonal matrices have orthogonal square roots
Definiteness of a general partitioned matrix $\mathbf M=\left$
Behavior of the spectral radius of a convergent matrix when some of the elements of the matrix change sign
Primitive Permutation Group with Subdegree 3
Solutions to $p+1=2n^2$ and $p^2+1=2m^2$ in Natural numbers.
Proving that $\frac{\csc\theta}{\cot\theta}-\frac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$
How to solve the equation $n^2 \equiv 0 \pmod{584}$?
What does “lower density” mean in this problem?
Domain of a function
Five Cubes in Dodecahedron
Calculating derivative by definition vs not by definition
Evaluate or simplify $\int\frac{1}{\ln x}\,dx$
What is the zero subscheme of a section
Two Steps away from the Hamilton Cycle

I have been trying to solve the following equation for months without much success. It has been so far a very frustrating endeavor.Please help.

Consider the diophantine equation: $x^2+y^2=z^r$ where $\gcd(x,y,z)=1$ and $r>2$.

Assuming that there exists a non-trivial triplet: $(x_0,y_0,z_0)$ satisfying: $$x_0^2+y_0^2=z_0^r$$ How do I find the parametrization of:$$ax^2+by^2=z^r$$?

- Find a example such $\frac{(x+y)^{x+y}(y+z)^{y+z}(x+z)^{x+z}}{x^{2x}y^{2y}z^{2z}}=2016$
- Golden Number Theory
- Finding a Pythagorean triple $a^2 + b^2 = c^2$ with $a+b+c=40$
- A diophantine equation with only “titanic” solutions
- Partition problem for consecutive $k$th powers with equal sums (another family)
- System of quadratic Diophantine equations

- Comparing $2013!$ and $1007^{2013}$
- Legendre's Proof (continued fractions) from Hardy's Book
- Find all integers satisfying $m^2=n_1^2+n_1n_2+n_2^2$
- Partition of ${1, 2, … , n}$ into subsets with equal sums.
- Is Legendre’s solution of the general quadratic equation the only one?
- Proof of Wolstenholme's theorem
- primegaps w.r.t. the m first primes / jacobsthal's function
- Is $2^k = 2013…$ for some $k$?
- Prove $2^b-1$ does not divide $2^a + 1$ for $a,b>2$
- Smallest positive element of $ \{ax + by: x,y \in \mathbb{Z}\}$ is $\gcd(a,b)$

It’s quite easy to parameterize,

$$\color{red}ax^2+\color{red}by^2 =z^k$$

**for odd $k$**. Assume,

$$x^2+by^2 = (p^2+bq^2)^k$$

$$(x+y\sqrt{-b})(x-y\sqrt{-b}) = (p+q\sqrt{-b})^k(p-q\sqrt{-b})^k$$

Equate factors and solve for $x,y$. Hence,

$$x =\frac{\alpha+\beta}{2},\quad y = \frac{\alpha-\beta}{2\sqrt{-b}},\quad\text{where}\quad \alpha = (p+q\sqrt{-b})^k,\quad \beta = (p-q\sqrt{-b})^k$$

For example, let $k = 3$. Then,

$$(p (p^2 – 3 b q^2))^2 + b(3 p^2q – b q^3)^2 = (p^2+b q^2)^3$$

Let $p = \sqrt{a}p$. Then,

$$\color{red}a(ap^2 – 3 b q^2)^2 + \color{red}b(3 ap^2q – b q^3)^2 = (ap^2+b q^2)^3\tag{k=3}$$

for ** free** variables $p,q$. Let $k = 5$,

$\color{red}a(a^2p^4 – 10 a b p^2q^2 + 5 b^2 q^4)^2 + \color{red}b(5 a^2p^4q – 10 a bp^2 q^3 + b^2 q^5)^2 = (ap^2+b q^2)^5\tag{k=5}$

and so on.

- Finding limit of function $f$, given limit of expression with $f(x)$.
- Splitting of the tangent bundle of a vector bundle
- Why does the absolute value disappear when taking $e^{\ln|x|}$
- Proof of the inequality $(x+y)^n\leq 2^{n-1}(x^n+y^n)$
- Connectedness of the spectrum of a tensor product.
- Why does a positive definite matrix defines a convex cone?
- Are there infinitely many primes and non primes of the form $10^n+1$?
- Functions defined by integrals (problem 10.23 from Apostol's Mathematical Analysis)
- How to show that $f : \mathbb{R}^n → \mathbb{R}^n$, $f(x) = \frac{h(\Vert x \Vert)}{\Vert x \Vert} x$, is a diffeomorphism onto the open unit ball?
- Connection between rank and matrix product
- A commutative ring $A$ is a field iff $A$ is a PID
- $(\mathbb{Z}\times\mathbb{Z})/\langle (0,1)\rangle$
- Are there $a,b \in \mathbb{N}$ that ${(\sum_{k=1}^n k)}^a = \sum_{k=1}^n k^b $ beside $2,3$
- Binomial theorem for non integers ? O_o ??
- Get location of vector/circle intersection?