All Questions
14 questions with no upvoted or accepted answers
12
votes
0
answers
704
views
Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?
Given an integer solution $s_m$ to the system,
$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$
$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$
and define the function,
$$F(s_m) = x_1+x_2+\dots+x_n$$
For $n\geq3$, using an ...
- 12.6k
10
votes
0
answers
258
views
Integral points on elliptic curve and the Lee norm
This question is based on small experiments I have done in Sagemath and if it is not research level, I will move it to MSE:
Let $E$ be an elliptic curve defined with coefficients in $\mathbb{Z}$.
The ...
user6671
7
votes
0
answers
237
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:
$a^2$ $b^2$ $c^2$
$ $ $ $ $ $ $ $ $ $ $d^2$
$e^2$ $f^2$ $g^2$
...
- 2,811
7
votes
0
answers
533
views
$a^5+b^5=c^5+d^5$ and polynomial identities
No nontrivial integer solutions to $$ a^5+b^5=c^5+d^5 \qquad (1)$$ are known.
(1) has infinitely many solutions in an extension of $\mathbb{Z}$
(root of $9-15x+37x^2 $ ) resulting
from a genus 0 ...
- 25.4k
3
votes
0
answers
176
views
For which primes $p$ in $\mathbb Z$ is $p\omega$ the sum of two cubes in $\mathbb Q(\omega)$?
This is related to an earlier question I posed —"Possible extensions of a conjecture …". Now that my note arXiv:2309.00162 has appeared I'll use it as a reference.
Elementary results(along ...
- 5,422
3
votes
0
answers
126
views
FLT and integral points on elliptic curves
For integers $x,y,z,t,n$ define $S_n : xy(x+y)=t^n$.
For $ n > 2$, Fermat's Last Theorem implies there are no integral
solution on $S_n$ with $x,y$ coprime and $xy(x+y) \ne 0$ since $x,y,x+y$ are
...
- 25.4k
3
votes
0
answers
326
views
Solving $(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-x_3)^{1/3} = {v}^{1/3}$ with elliptic curves
Let $x_1$,$x_2$,$x_3$ be the roots of the cubic $x^3+px+q$ over $\mathbb Q$, the idea is that rational solutions $(u,v)$ of the equation
$$(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-x_3)^{1/3} = {v}^{1/3} \...
- 205
3
votes
0
answers
578
views
On Choudhry's $x_1^k+x_2^k+x_3^k+x_4^k = y_1^k+y_2^k+y_3^k+y_4^k$ for $k=7$?
I. Fifth Powers
The Diophantine equation,
$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$
for $k=5$ is quite well-explored. It has an infinite number of primitive solutions (as points on an elliptic ...
- 12.6k
2
votes
0
answers
52
views
Infinitely many coprime solutions of $F(x,y)= k(a_1 x + a_2 y)^2 z^2$?
This might be related to an open problem.
Let $F(x,y)$ be homogeneous degree 4 squarefree polynomial
with integer coefficients and
$h(x,y)=a_1 x + a_2 y$ and $\gcd(F,h)=1$ and $k$ be integer.
Consider ...
- 25.4k
2
votes
0
answers
171
views
trivial solutions for Diophantine equations
Let $K$ be an odd degree number field. Consider the Diophantine equation:
$$
X^4 + bY^4 =Z^2
$$
where $b\neq 0$.
Say we know that the above equation has only trivial roots in $K$ (for some fixed ...
- 1,283
1
vote
0
answers
274
views
4-distance problem and elliptic curves
The 4-distance problem is an open question(as far as I know it is still open) that asks if there exists a point P on the Euclidean plane such that its distances to all four points of a unit square are ...
- 547
1
vote
0
answers
146
views
On $x^4+16z^n=y^2$ and $x^4+z^n=y^2$
For $n>4$ and coprime integers $x,y,z$ consider the diophantine equations:
$$x^4+16z^n=y^2 \qquad (1)$$
and
$$x^4+z^n=y^2 \qquad (2)$$.
(2) is special case of Fermat Catalan and is solved.
For ...
- 25.4k
0
votes
0
answers
197
views
On the integer solutions of the equation $y^2 = x^3 + n$
Let $n$ be a nonzero integer. I am interested in the integer solutions $(x, y)$ to the equation $y^2 = x^3 + n$.
Let $S$ be the set of all integer solutions $(x, y)$ to this equation.
I am wondering ...
- 95
0
votes
0
answers
178
views
Elementary method for finding integer solutions for certain types of elliptic curve
There are some problems in high school Olympiad that ask to find integer solutions of the form $Q(x^2) = dy^2 (*)$ where $Q$ is a quadratic polynomial and $d$ is an absolute constant and quite often, $...
- 193