# Tagged Questions

1answer
87 views

### Diophantine equation: $2 a^2 + 2 b^2 = c^2 + d^2$ [duplicate]

I am looking for integer solutions of the following equation: $$2a^{2} + 2b^{2} - c^{2} - d^{2} = 0$$ Preferentially the solutions should obey $a+b+c+d=0$. By inspection I found the solutions: ...
1answer
33 views

### Positive Integer points of $f(x)=\frac{1}{c-\frac{1}{x}}$, where c is fixed

So I am looking for the integer solutions of $f(x)=\frac{1}{c-\frac{1}{x}}$ for fixed $c\in \mathbb{Q}$ i.e. points $(x,f(x))\in \mathbb{N}\times \mathbb{N}$. (The c equals $\frac{4}{n}-\frac{1}{k}$ ...
1answer
103 views

### What's the approach to find out if this equation has integer solutions?

The equation is $u^3(s^4 + (r-1)^4) - s^4(t - 1)^3 = 0$ Has no integer solutions for $u,s \neq 0$. How do mathematicians today approach this problem? Sorry for broadness, just looking for a ...
0answers
111 views

### General quadratic diophantine equation.

Here is my problem: I am given a general quadratic diophantine equation: $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$ where $x$ and $y$ are variables with integers $a,b,c,d,e,f$. I have to show that if the ...
1answer
145 views

### sum of three cubes and parametric solutions

The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$. In other words, there are finite many polynomial triples ...
0answers
165 views

### Diophantine equations/Diophantine Geometry

I am very knew to this site and I am eagerly waiting for solutions of: (1) Let $x$ be an algebraic number with degree $n > 1$. Then there exists only finitely many rational numbers $p/q$ (in ...
2answers
175 views

### Does this equation have integer solutions

Let $g\geq 2$ be an integer. (It will be the genus of some curve.) Are there positive integers $d$ and $e$ such that the equality $$(e-2)(e-1) = 2d(g-1)+2$$ holds?
2answers
351 views

### Generating Pythagorean triples for $a^2+b^2=5c^2$?

Just trying to figure out a way to generate triples for $a^2+b^2=5c^2$. The wiki article shows how it is done for $a^2+b^2=c^2$ but I am not sure how to extrapolate.
0answers
24 views

### Can one determine in finite time whether a point is $S$-integral

Let $x$ be a $\mathbf{Q}$-rational point of $\mathbf{P}^1-\{0,1,\infty\}$. Let $S$ be a finite set of primes. How do I check in finite time whether $x$ is $S$-integral or not? I know how to do this ...
1answer
121 views

### Diophantine equations and Groebner bases

I'm trying to teach myself the basics of algebraic geometry and have run into something that I don't understand. I know that the problem of deciding whether a Diophantine equation $P(\vec{x}) = 0$ ...
1answer
486 views