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There are three cases when $1/D+1/E+1/F=1$, which give $(D,E,F)=(2,3,6)$, $(4,4,2)$, and $(3,3,3)$. In these three cases the equations define curves that are in fact elliptic curves of rank $0$, and t …

You don't need any conjectures for this. If two curves have infinitely many points in common then they have a common component (this is a basic fact of the Zariski topology). As $x^2+xy-y^2-1$ and $x^ …

Given the square roots of $1$ modulo $N$ you can deduce the square roots of $1$ modulo $N^2$ just by using Hensel's Lemma (without factoring). Specifically let $r$ be one of the square roots of $1$ mo …

This a theorem of Hermite (true for number fields of any degree). I don't know of a reference, but I think it is good to look at the theorems of Hunter and Martinet which give you a finite search regi …

Let $n \ge 2$. Let $p_1,\dots,p_m$ be distinct primes $\equiv 1 \pmod{n}$. Let $N=p_1 p_2 \cdots p_m$. If $\gcd(a,N)=1$ and the equation $x^n \equiv a \pmod{N}$ has a solution then it has $n^m$ soluti …

The $n$-th Mersenne number is $M_n=2^n-1$. Write $M_n=a_n b_n^2$ where $a_n$ is positive and squarefree. Question 1: What lower bound can be proved for $a_n$? Let $A$ be the set of all possible $a_ …

There's an elementary way of solving this (and similar equations). Let's start with $$ 1+2^n=5^a(1+10^m) $$ which you want to solve in positive integers $n$, $a$, $m$. Clearly $m < n$ and $a < n$. As …

I'm not sure whether the second question is inadvertently misworded. As it is, the answer can easily be shown to be no. Fix $h$ and let $A$ be huge compared to $h$. Let $$ f(x,y)=A(x-y)(x-2y)\cdots (x …

What you need is the theory of lower bounds for linear forms in logarithms. A good place to start reading about this is the following article by Evertse: www.math.leidenuniv.nl/~evertse/dio2011-linfo …

There is a standard method for computing all integral points on an elliptic curve using David's bounds and lattice reduction. The method can be found in the book: Nigel Smart, "The Algorithmic Resolut …

This kind of problem usually requires a little algebraic number theory. Joe Silverman sketches one possible approach in the comments. Here is another. Let's rewrite as $$ (2y)^2-5^n=-1. $$ If $n$ is e …

Hi, There is no claim in my cv or elsewhere that me and Sander have solved the equation x^3+y^5+z^7=0. All my cv claims is that we're writing a paper on it! That's not the same thing. All the best, …

Although Junkie has answered the question, I'd like to point out that in the case of parametrized families of elliptic curves (such as this) it is often easy to find an explicit subfamily with positiv …

Are there absolute constants $N$ and $B$ such that the following is true? Let $p>B$ be a prime. Let $q(x_0,\dotsc,x_n)$ and $c(x_0,\dotsc,x_n)$ be homogeneous of degree $2$ and $3$ with coefficients …

I guess this answer complements Gene's answer above. Here is an example to think about. Let $$ A=(x^2-2)(x^2-17)(x^2-34). $$ It's an easy exercise in quadratic reciprocity to show that $\mathcal{P}(A) …