1
vote
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 $\frac{4}{n}-\frac{1}{k}$ ...
1
vote
0answers
23 views

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 ...
6
votes
1answer
211 views

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 ...
5
votes
1answer
240 views

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 ...
1
vote
1answer
50 views

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 ...
5
votes
3answers
396 views

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 ...
3
votes
1answer
89 views

Pell-like equations and continued fractions

Why does the continued fraction method work? Could be applied in order to solve, for example, $x^{17}-19y^{17}=1$ ?
5
votes
3answers
193 views

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 ...
4
votes
2answers
130 views

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 ...
13
votes
2answers
260 views

How did Letac solve $x_1^k + x_2^k + \dots +x_9^k = 0$ for $k = 1, 3, 5, 7$ in 1942?

It's quite easy to find integer solutions to, $$x_0^k + x_1^k + \dots +x_9^k = 0$$ for $k = 1, 3, 5, 7$. One I found is, if $x^2-10y^2 = 9$, then, $$1 + 5^k + (3+2y)^k + (3-2y)^k + (-3+3y)^k + ...
23
votes
1answer
1k views

How to compute rational or integer points on elliptic curves

This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever ...
1
vote
1answer
102 views

Diophantine Equation $x^n+y^n=z^n$

Problem Using simple mathematical operators (+,- ,> etc.) can it be shown that (assuming $ x<y$) Fermat’s theorem is always true when $$ n\ge x$$ Request I am sure this approach has been ...
10
votes
1answer
192 views

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 $x^y-y^x=11$
8
votes
4answers
349 views

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 ...
1
vote
0answers
190 views

Quadratic fields and solving Diophantine equations

I would like to learn to solve Diophantine equations and I think my next step would be quadratic fields or number fields. What are kind of methods there are to use those on solving equations? And what ...
3
votes
0answers
250 views

Counting Solutions of Diophantine Inequalities

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 ...