All Questions
Tagged with diophantine-equations elliptic-curves
78 questions
5
votes
2
answers
365
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Are there integer solutions of $m^4+m^2n^2+n^4=k^2$?
The recent question about Sets of integers with same sum and same sum of reciprocals, with its wealth of solutions, raises naturally the question: Can we require all those numbers to be squares? Of ...
- 22.5k
7
votes
1
answer
625
views
Action of the symmetric group $S_3$ on an elliptic curve $E$ defined over $\mathbb{Z}$
I came up with the following question on a facebook group: find the positive integer solutions of the equation $$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$$
Now clearly this is very difficult, ...
- 2,858
-5
votes
1
answer
150
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On Mordell equation $y^2=x^3+k$ [closed]
Have the Mordell equation $y^2=x^3+k$ solved for all integer $k$ or not?
Please Could you tell me about a good review papers about such equation.
- 17
18
votes
2
answers
2k
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What is the taxicab number for rational fourth powers?
The taxicab number is the smallest integer that can be expressed as a sum of two positive integer cubes in two different ways, and it is equal to $1729=12^3+1^3=10^3+9^3$. There are generalizations to ...
- 39.9k
3
votes
1
answer
222
views
Large integral points on the quadratic twist $ D y^2=x^3+A x +B$
For integers $A,B,D$ and $D$ squarefree let $E : y^2=x^3+A x + B$
and $E_D$ be the quadratic twist of the elliptic curve $E$:
$$ E_D : D y^2=x^3+Ax +B$$
$E_D$ is isomorphic to $ E'_D : y^2=x^3+D^2 A ...
CommunityBot
- 1
2
votes
0
answers
52
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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
14
votes
1
answer
408
views
Can you "slice" a triangular number into three equal slices?
Problem statement:
Does there exist positive integers $a<b<c$ such that
$$1 + 2 + \dots + (a-1) = (a+1) + \dots + (b-1) = (b+1) + \dots + c?$$
(Note that $a$ and $b$ are not in the sums.)
...
- 12.9k
0
votes
0
answers
197
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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 ...
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6
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2
answers
482
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Difficult elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?
Similar to the case $x^4+y^4+z^4 = 1$ discussed in this MO post, define the system,
$$x^4+y^4+z^4+1 = (x+y+z+1)^4\tag1$$
$$\frac{x^2+x+1}{(x+y+1)(x+z+1)}=u\tag2$$
$$\frac{y^2+y+1}{(y+z+1)(y+x+1)}=v\...
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7
votes
1
answer
463
views
A parametric elliptic curve for $x^4+y^4+z^4 = 1$?
Noam Elkies found that $x^4+y^4+z^4 = 1$ has infinitely many rational points $xyz \neq 0$ using an elliptic curve. We use a different approach that will produce pairs of solutions and a parametric ...
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13
votes
2
answers
1k
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On Euler's elliptic curve for $A^4+B^4 = C^4+D^4$?
To solve,
$$A^4+B^4 = C^4+D^4$$
we use Euler's method. Let,
$$(p+q)^4+(r-s)^4=(p-q)^4+(r+s)^4$$
and define $p = (a^3 - b),\, q = a y,\, r = b (a^3 - b),\, s = y.\,$ The equation above transforms to ...
- 1,986
3
votes
0
answers
176
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For which primes $p$ in $\mathbb Z$ is $p\omega$ the sum of two cubes in $\mathbb Q(\omega)$?
This is related to an earlier question I posed —"Possible extensions of a conjecture …". Now that my note arXiv:2309.00162 has appeared I'll use it as a reference.
Elementary results(along ...
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0
votes
0
answers
178
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Elementary method for finding integer solutions for certain types of elliptic curve
There are some problems in high school Olympiad that ask to find integer solutions of the form $Q(x^2) = dy^2 (*)$ where $Q$ is a quadratic polynomial and $d$ is an absolute constant and quite often, $...
- 39.9k
6
votes
1
answer
964
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The elliptic curve for $x_1^9+x_2^9+\dots+x_6^9 = y_1^9+y_2^9+\dots+y_6^9$
I. Theorem: "If there are $a,b,c,d,e,f$ such that,
$$a+b+c = d+e+f\tag1$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag2$$
$$3u^3-3uv+w=-def\qquad\tag3$$
with $(u,v,w)$ as the symmetric polynomials $u=a+b+c,\; ...
- 12.6k
0
votes
1
answer
107
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Is there a method to check if two sections of an elliptic surface are dependent over the endomorphism ring or not?
Let $E\!: y^2 = x^3 + a(t)x + b(t)$ be an elliptic curve over the function field $\mathbb{F}_{q}(t)$ over a finite field $\mathbb{F}_{q}$ of characteristic $5$ or greater. For simplicity, let $E$ be ...
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1
vote
1
answer
262
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On the equation $a^4+b^4+c^4=2d^4$ in natural numbers with $a<b<c<d$
I asked a simillar question with the weaker restriction:
On the equation $a^4+b^4+c^4=2d^4$ in positive integers $a\lt b\lt c$ such that $a+b\ne c$
.
I couldn't find any solution to this equation. ...
CommunityBot
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3
votes
0
answers
578
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On Choudhry's $x_1^k+x_2^k+x_3^k+x_4^k = y_1^k+y_2^k+y_3^k+y_4^k$ for $k=7$?
I. Fifth Powers
The Diophantine equation,
$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$
for $k=5$ is quite well-explored. It has an infinite number of primitive solutions (as points on an elliptic ...
- 12.6k
175
votes
2
answers
66k
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Estimating the size of solutions of a diophantine equation
A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ?
(I do not know any such numbers).
B. Suppose that $\frac{a}{b+c} + \...
- 111
11
votes
2
answers
1k
views
Sum of consecutive cubes
I'm investigating when the sum of $n$ consecutive cubes equals a cube, i.e., for which $n$ does
$$\sum_{i=0}^{n-1} (k+i)^3 = k^3 + (k+1)^3 + \cdots + (k+n-1)^3 = Y^3 $$
have nontrivial solutions $(k,...
- 1,446
40
votes
2
answers
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$x^4+y^4$ powerful for relatively prime $x,y$
I asked this question on the NMBRTHRY mailing list on
17 February 2014, but it remains unsolved as far as I know.
Recall that a "powerful
number" is a positive integer whose prime ...
- 4,713
5
votes
1
answer
526
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Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$
According to a conjecture there are no three
consecutive powerful numbers.
Necessary condition for this is integer solution of
$$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$
What are integer solutions of ...
- 801
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
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$
...
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1
vote
0
answers
143
views
A specific Diophantine equation related to the congruent number question
Let $n$ be an odd square free natural number. J.B. Tunnel in his 1983 paper, showed that a number $n$ is congruent, if and only if the number of triples of integers satisfying $2x^2+y^2+8z^2=n$ is ...
- 159
1
vote
2
answers
435
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On the equation $x^3 + y^3 =cz^3$
What are the characteristics of the values of $c$ for which the equation $x^3 + y^3 = cz^3$ has pairwise coprime non-zero integral solutions where $x \neq \pm y$ ? For instance, it is known that $c$ ...
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1
answer
205
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Are there orthogonal Cauchy-like matrices with rational entries for any given size?
This is inspired by a recent question about the existence of orthogonal Cauchy-like matrices. It is proved that there are indeed such matrices, i.e. there are vectors $x,y,r,s\in\mathbb R^n$ such ...
CommunityBot
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8
votes
4
answers
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Status of $x^3+y^3+z^3=6xyz$
In
Erik Dofs, Solutions of $x^3 + y^3 + z^3 = nxyz$, Acta Arithmetica 73 (1995) pp. 201–213, doi:10.4064/aa-73-3-201-213, EuDML
the author has studied the Diophantine equation
\begin{equation}
x^3+y^...
- 1,567
26
votes
2
answers
1k
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Why do the $2$-Selmer ranks of $y^2 = x^3 + p^3 $ and $y^2 = x^3 - p^3 $ agree?
I was playing around with sage, when I found that the Mordell-Weil ranks (over $\mathbb{Q}$) of the elliptic curves $y^2=x^3+p^3$ and $y^2=x^3-p^3 $ almost always agree, for $p$ prime. The first few ...
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32
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4
answers
5k
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Why is this "the first elliptic curve in nature"?
The LMFDB describes the elliptic curve 11a3 (or 11.a3) as "The first elliptic curve in nature". It has minimal Weierstraß equation
$$
y^2 + y = x^3 - x^2.
$$
My guess is that there is some ...
- 35.5k
10
votes
0
answers
259
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 ...
CommunityBot
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0
votes
1
answer
325
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On the elliptic curve $y^2 = x^3 + z^{4k}$
Are there any rational numbers $x, y, z$ with $xyz \neq 0$ such that $y^2 = x^3 + z^{4k}$ for some $k \in \mathbb{Z}_{>1}$ ?
- 47.4k
13
votes
1
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499
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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 ...
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2
answers
608
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Rational points on the elliptic curve $y^2 = x^{3} - t^{2}z^3$
What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.
ADDENDUM 1. I have just noticed that if $z^3 ...
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4
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1
answer
564
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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 ...
- 1,097
16
votes
2
answers
410
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$3$-ranks of elliptic curves and representations $p=ax^3+by^3$
Let $p$ be a prime with $p\equiv2\pmod3$ and $E_p$ the elliptic curve $y^2=x^3+9p^2$
which has a rational $3$-torsion point. Let $\alpha$ from $E_p(\mathbb Q)$ to $\mathbb Q^*/{\mathbb Q^*}^3$ be the $...
- 1,986
3
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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
8
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2
answers
643
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Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers
I am interested to know if a similar theorem that shows this answer of the post
Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
6
votes
3
answers
1k
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Are there Heronian triangles that can be decomposed into three smaller ones?
Is there anything known about the existence of Heronian triangles ABC (i.e. with rational side lengths and rational area) that can be decomposed into three Heronian triangles ABD, BCD, CAD? ...
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2
votes
1
answer
573
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Integer points of one Mordell equation
How can I determine all integer points of the following equation
$$y^2=x^3+10546$$
I tried Magma with
IntegralPoints(EllipticCurve([0,10546]));
but got the ...
- 8,945
9
votes
3
answers
531
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Diophantine equations $ax^4+by^2=c$ in rational numbers
Are there general ways for given rational coefficients $a,b,c$ (I am particularly interested in $a=3,b=1,c=8076$, but in general case too) to answer whether this equation has a rational solution or ...
- 1
7
votes
4
answers
884
views
Extending rational Diophantine triples to sextuples
(This is a follow-up to a previous post.) A rational Diophantine $m$-tuple is a set of rationals {$a_1,a_2,\dots a_m$} such that (with $i\neq j$), all $a_i a_j+1$ is a square. Problem: Find a class of ...
- 12.6k
18
votes
3
answers
2k
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More elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?
(Note: See also the $a^4+b^4+c^4 = 1$ version in this old MSE post.)
The equation discussed in a paper by Jacobi and Madden,
$$a^4+b^4+c^4+d^4 = (a+b+c+d)^4 = z^4\tag1$$
or equivalently,
$$(p-2q + ...
- 205
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
$$(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-x_3)^{1/3} = {v}^{1/3} \...
CommunityBot
- 1
4
votes
1
answer
352
views
Squares in the set $\{\sum_{j=1}^m j^2: m\in\mathbb{N}\}$ [closed]
Are there infinitely many squares in the set $$\{\sum_{j=1}^m j^2: m\in\mathbb{N}\} ?$$
- 34.3k
3
votes
1
answer
246
views
Fermat's cubic equation in quadratic extension of $\mathbb{Q}$
Is still relevant or interesting be capable to bring a criteria in order to classifly quadratic extensions of $\mathbb{Q}$ based on the existence or not existence of non-trivial solutions of Fermat's ...
- 50.6k
2
votes
0
answers
171
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trivial solutions for Diophantine equations
Let $K$ be an odd degree number field. Consider the Diophantine equation:
$$
X^4 + bY^4 =Z^2
$$
where $b\neq 0$.
Say we know that the above equation has only trivial roots in $K$ (for some fixed ...
- 1,283
8
votes
1
answer
206
views
Integral complete 4-partite graphs
For given block sizes $a<b<c<d$, consider the complete 4-partite graph $K_{a,b,c,d }$.
Can such a graph be integral, i.e. have only integer eigenvalues?
It is easy to see that the ...
- 79.9k
1
vote
0
answers
146
views
On $x^4+16z^n=y^2$ and $x^4+z^n=y^2$
For $n>4$ and coprime integers $x,y,z$ consider the diophantine equations:
$$x^4+16z^n=y^2 \qquad (1)$$
and
$$x^4+z^n=y^2 \qquad (2)$$.
(2) is special case of Fermat Catalan and is solved.
For ...
- 25.4k
7
votes
0
answers
533
views
$a^5+b^5=c^5+d^5$ and polynomial identities
No nontrivial integer solutions to $$ 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 ...
- 13.4k
40
votes
1
answer
2k
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Diophantine equation for 2016: triangular $|{\rm GL}_2({\bf F}_q)|$
For a prime power $q$ the group ${\rm GL}_2({\bf F}_q)$ has
$(q^2-1)(q^2-q)$ elements. This happens to be a triangular number for
$q=2$ (being $6 = 1+2+3$), and $-$ more notably, especially this year ...
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