Search Results
Search type | Search syntax |
---|---|
Tags | [tag] |
Exact | "words here" |
Author |
user:1234 user:me (yours) |
Score |
score:3 (3+) score:0 (none) |
Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
Views | views:250 |
Code | code:"if (foo != bar)" |
Sections |
title:apples body:"apples oranges" |
URL | url:"*.example.com" |
Saves | in:saves |
Status |
closed:yes duplicate:no migrated:no wiki:no |
Types |
is:question is:answer |
Exclude |
-[tag] -apples |
For more details on advanced search visit our help page |
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
3
votes
0
answers
171
views
Proof of when 3 is a cubic residue modulo primes
I have recently been learning about cubic characters, and the machinery of Gauss and Jacobi sums used to prove the cubic reciprocity theorem, and using this, I can now determine when any prime is a cu …
5
votes
3
answers
286
views
Small Galois group solution to Fermat quintic
I have been looking into the Fermat quintic equation $a^5+b^5+c^5+d^5=0$. To exclude the trivial cases (e.g. $c=-a,d=-b$), I will take $a+b+c+d$ to be nonzero for the rest of the question. It can be s …
7
votes
0
answers
235
views
Magic hourglass of squares hyperelliptic equation
I have been looking into the problem of the magic square of squares, or more specifically, the magic hourglass of squares, like so:
$a^2$ $b^2$ $c^2$
$ $ $ $ $ $ $ $ $ $ $d^2$
$e^2$ $f^2$ $g^2$
Wh …
13
votes
1
answer
497
views
On the equation $a^6+b^6+c^6=d^2$
I have been studying the equation $a^6+b^6+c^6=d^2$, trying to find rational solutions. I know it is a K3 surface, with high Picard rank, so there should be rational or elliptic curves on it.
When Elk …
10
votes
5
answers
763
views
Reference request: Diophantine equations
I am looking for a textbook, or preferably lectures, on the subject of Diophantine equations. I am familiar with the basic principles of modular arithmetic, conics and the Hasse Principle, and the bas …
0
votes
0
answers
203
views
On Sums of powers II
In my previous question, I asked about nontrivial sums of four powers $a^m+b^n+c^k=d^l$, and whether the nature of the solutions depend on whether $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}+\frac{1}{l}$ is …
4
votes
1
answer
497
views
On sums of powers
I was considering the Fermat Catalan conjecture, where the equation $a^m+b^n=c^k$ has only finitely many nontrivial solutions (with coprime $a, b, c$) with $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}=1$ (and …
1
vote
Rational points on the elliptic curve $y^2 = x^{3} - t^{2}z^3$
If we rewrite the equation as $y^2+t^{2}z^3 = x^3$, and let $v=tz$, then $y^2+zv^2=x^3$.
If you factorize over $Q[\sqrt{-z}]$, then $(y+v\sqrt{-z})(y-v\sqrt{-z})=x^3 $.
Let $x=(a+b\sqrt{-z})(a-b\sqrt{ …
4
votes
1
answer
538
views
The number of perfect squares which can occur in an arithmetic progression of length n
This is a similar question to https://math.stackexchange.com/questions/2023399/the-maximum-number-of-perfect-squares-that-can-be-in-an-arithmetic-progression/3693487#3693487
Let f(n) be the maximum n …
4
votes
1
answer
406
views
Solutions to diophantine equation
I have been working on solutions to $x^5+y^5+z^5=1$, and I found that the three solutions of $x^3+bx+\frac{1}{5b}$ satisfy that equation. Multiplying by $5b$: $5xb^2+5x^3b+1=0$, then solving for b yie …
2
votes
Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$
I found a parameterized solution of $X^5+Y^5+Z^5=T^5$ with $X$ and $T$ rational and $Y$ and $Z$ complex rational:
$(k^2-4k+1)^5+(2k-2+(k^2-2k+3)i)^5+(2k-2-(k^2-2k+3)i)^5=(k^2-3)^5$
There is an almos …
-1
votes
1
answer
240
views
examples of non-unique factorisation in cyclotomic fields [closed]
I was looking into cyclotomic extensions of the natural numbers, and I found that extending the naturals with the 23rd root of unity caused the ring to no longer be a UFD. In other words, there now ex …
12
votes
1
answer
3k
views
Factorization of polynomials in two variables
I have read, from the question
Irreducibility of polynomials in two variables, that all polynomials $f(x)-g(y)$, where $f, g$ are indecomposable polynomials, and there are no $a, b$ such that $g(ax+b …
0
votes
1
answer
683
views
Solutions to x^2+n=y^3 diophantine equation [closed]
I recently read http://cp4space.wordpress.com/2013/11/25/crash-course-in-gaussian-integers/, where it teaches you how to solve some diophantine equations of the form x^2+n=y^3 using the gaussian integ …