# Tagged Questions

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### Superelliptic curves

I'm trying to find information on superelliptic curves and how to solve them over the integers. The equation is $$y^k = f(x)$$ where $k=3$ and $f$ has degree $d=3$. Does anyone know any ...
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### 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 $\frac{4}{n}-\frac{1}{k}$ ...
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### Integral points on varieties and solutions to Diophantine equations

I am looking for a book (or article, or notes...) explaining details about the link between integral points on varieties defined as complement of certain divisors and integral solutions to the ...
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### Are differences between powers of 2 equal to differences between powers of 3 infinitely often?

Consider the equation $2^a-2^b=3^c-3^d$ where $a>b>0$, $c>d>0$, and $a,b,c,d$ are all integers. A computer search for solutions with $a,c\le20$ only finds 8-2=9-3, 32-8=27-3, and ...
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### On Bachet's Duplication Formula and the number $-432$

While reading "Rational Points on Elliptic Curves" by Silverman and Tate, I came across this interesting passage about Bachet's duplication formula: I know how to derive Bachet's duplication ...
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### two questions about Diophantine Equation

I am reading an article Modular Arithmetic by Richard Taylor. I have 2 questions: For which $n$, $x^2+y^2=nz^2$ has nontravial solutions? What are the solutions? A beautiful theorem of Hermann ...
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### Erdös-Straus conjecture

I'm reading a lot about the Erdös-Straus Conjecture (ESC), a conjecture that states that for every natural number $p \geq 2$, there exists a set of natural numbers $a, b, c$ , such that the following ...
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### Pell-like equations and continued fractions

Why does the continued fraction method work? Could be applied in order to solve, for example, $x^{17}-19y^{17}=1$ ?
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### Sums of powers being powers of the sum

I'm looking for literature on solving problems of the form $$n_1^\alpha+\cdots+n_k^\alpha=(n_1+\cdots+n_k)^\beta$$ for positive integers $n_1,\ldots,n_k$ and fixed parameters $k$ and ...
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### Foundation on Diophantine Analysis and Number Theory

I want to read particularly about diophantine Analysis and Elementary Number Theory from a novice level. The books which I found on net: A Guide to Elementary Number Theory by Underwood Dudley ...
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### Diophantine equation $x^y-y^x=11$

How can one find all integer solutions to $x^y-y^x=k$, for a given k? Example case $x^y-y^x=11$
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### Is there a catalogue of solved Diophantine equations?

Is there a book, website or something else aiming to catalogue all or many of the Diophantine equations that have already been solved? I have two tiny books by Sierpiński in which he gives some of ...
I understand that Diophantine Analysis is an enormous field! Without first determining the solution set, suppose I'd like to calculate the number of non-negative integer solutions $(x,y,z)$ of ...