All Questions
71 questions
5
votes
2
answers
363
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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
-5
votes
1
answer
150
views
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
views
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
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
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
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
6
votes
2
answers
482
views
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,
$$x^4+y^4+z^4+1 = (x+y+z+1)^4\tag1$$
$$\frac{x^2+x+1}{(x+y+1)(x+z+1)}=u\tag2$$
$$\frac{y^2+y+1}{(y+z+1)(y+x+1)}=v\...
- 1,986
7
votes
1
answer
462
views
A parametric elliptic curve for $x^4+y^4+z^4 = 1$?
Noam Elkies found that $x^4+y^4+z^4 = 1$ has infinitely many rational points $xyz \neq 0$ using an elliptic curve. We use a different approach that will produce pairs of solutions and a parametric ...
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13
votes
2
answers
1k
views
On Euler's elliptic curve for $A^4+B^4 = C^4+D^4$?
To solve,
$$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$$
and define $p = (a^3 - b),\, q = a y,\, r = b (a^3 - b),\, s = y.\,$ The equation above transforms to ...
- 1,986
3
votes
0
answers
176
views
For which primes $p$ in $\mathbb Z$ is $p\omega$ the sum of two cubes in $\mathbb Q(\omega)$?
This is related to an earlier question I posed —"Possible extensions of a conjecture …". Now that my note arXiv:2309.00162 has appeared I'll use it as a reference.
Elementary results(along ...
- 2,031
0
votes
0
answers
178
views
Elementary method for finding integer solutions for certain types of elliptic curve
There are some problems in high school Olympiad that ask to find integer solutions of the form $Q(x^2) = dy^2 (*)$ where $Q$ is a quadratic polynomial and $d$ is an absolute constant and quite often, $...
- 39.9k
6
votes
1
answer
964
views
The elliptic curve for $x_1^9+x_2^9+\dots+x_6^9 = y_1^9+y_2^9+\dots+y_6^9$
I. Theorem: "If there are $a,b,c,d,e,f$ such that,
$$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$$
with $(u,v,w)$ as the symmetric polynomials $u=a+b+c,\; ...
- 12.6k
1
vote
1
answer
262
views
On the equation $a^4+b^4+c^4=2d^4$ in natural numbers with $a<b<c<d$
I asked a simillar question with the weaker restriction:
On the equation $a^4+b^4+c^4=2d^4$ in positive integers $a\lt b\lt c$ such that $a+b\ne c$
.
I couldn't find any solution to this equation. ...
CommunityBot
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3
votes
0
answers
578
views
On Choudhry's $x_1^k+x_2^k+x_3^k+x_4^k = y_1^k+y_2^k+y_3^k+y_4^k$ for $k=7$?
I. Fifth Powers
The Diophantine equation,
$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$
for $k=5$ is quite well-explored. It has an infinite number of primitive solutions (as points on an elliptic ...
- 12.6k
175
votes
2
answers
66k
views
Estimating the size of solutions of a diophantine equation
A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ?
(I do not know any such numbers).
B. Suppose that $\frac{a}{b+c} + \...
- 111
11
votes
2
answers
1k
views
Sum of consecutive cubes
I'm investigating when the sum of $n$ consecutive cubes equals a cube, i.e., for which $n$ does
$$\sum_{i=0}^{n-1} (k+i)^3 = k^3 + (k+1)^3 + \cdots + (k+n-1)^3 = Y^3 $$
have nontrivial solutions $(k,...
- 1,446
40
votes
2
answers
3k
views
$x^4+y^4$ powerful for relatively prime $x,y$
I asked this question on the NMBRTHRY mailing list on
17 February 2014, but it remains unsolved as far as I know.
Recall that a "powerful
number" is a positive integer whose prime ...
- 4,713
5
votes
1
answer
526
views
Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$
According to a conjecture there are no three
consecutive powerful numbers.
Necessary condition for this is integer solution of
$$ 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
2
answers
435
views
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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8
votes
4
answers
2k
views
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
views
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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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 ...
CommunityBot
- 1
0
votes
1
answer
325
views
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
answer
499
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 ...
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-3
votes
2
answers
608
views
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
votes
1
answer
563
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 ...
- 1,097
16
votes
2
answers
410
views
$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
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
8
votes
2
answers
643
views
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
views
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
views
Integer points of one Mordell equation
How can I determine all integer points of the following equation
$$y^2=x^3+10546$$
I tried Magma with
IntegralPoints(EllipticCurve([0,10546]));
but got the ...
- 8,945
9
votes
3
answers
531
views
Diophantine equations $ax^4+by^2=c$ in rational numbers
Are there general ways for given rational coefficients $a,b,c$ (I am particularly interested in $a=3,b=1,c=8076$, but in general case too) to answer whether this equation has a rational solution or ...
- 1
7
votes
4
answers
884
views
Extending rational Diophantine triples to sextuples
(This is a follow-up to a previous post.) A rational Diophantine $m$-tuple is a set of rationals {$a_1,a_2,\dots a_m$} such that (with $i\neq j$), all $a_i a_j+1$ is a square. Problem: Find a class of ...
- 12.6k
18
votes
3
answers
2k
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More elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?
(Note: See also the $a^4+b^4+c^4 = 1$ version in this old MSE post.)
The equation discussed in a paper by Jacobi and Madden,
$$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}\} ?$$
- 34.3k
3
votes
1
answer
246
views
Fermat's cubic equation in quadratic extension of $\mathbb{Q}$
Is still relevant or interesting be capable to bring a criteria in order to classifly quadratic extensions of $\mathbb{Q}$ based on the existence or not existence of non-trivial solutions of Fermat's ...
- 50.6k
2
votes
0
answers
171
views
trivial solutions for Diophantine equations
Let $K$ be an odd degree number field. Consider the Diophantine equation:
$$
X^4 + bY^4 =Z^2
$$
where $b\neq 0$.
Say we know that the above equation has only trivial roots in $K$ (for some fixed ...
- 1,283
8
votes
1
answer
206
views
Integral complete 4-partite graphs
For given block sizes $a<b<c<d$, consider the complete 4-partite graph $K_{a,b,c,d }$.
Can such a graph be integral, i.e. have only integer eigenvalues?
It is easy to see that the ...
- 79.9k
1
vote
0
answers
146
views
On $x^4+16z^n=y^2$ and $x^4+z^n=y^2$
For $n>4$ and coprime integers $x,y,z$ consider the diophantine equations:
$$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)$$ are known.
(1) has infinitely many solutions in an extension of $\mathbb{Z}$
(root of $9-15x+37x^2 $ ) resulting
from a genus 0 ...
- 13.4k
40
votes
1
answer
2k
views
Diophantine equation for 2016: triangular $|{\rm GL}_2({\bf F}_q)|$
For a prime power $q$ the group ${\rm GL}_2({\bf F}_q)$ has
$(q^2-1)(q^2-q)$ elements. This happens to be a triangular number for
$q=2$ (being $6 = 1+2+3$), and $-$ more notably, especially this year ...
- 602
3
votes
1
answer
164
views
Existence of Pillai equations with Catalan type solutions?
In Catalan's conjecture we have $$x^m-y^n=1$$ having solution $(3,2,1,1)$ and $(3,2,2,3)$.
Call $$ax^m-by^n=k$$ to be Pillai Diophantine equation.
Is it true no Pillai Diophantine equation exists ...
CommunityBot
- 1
8
votes
2
answers
729
views
An elliptic curve for Ramanujan-type cubic identities?
Given the roots $x_i$ of the depressed cubic,
$$x^3+px+q=0$$
with rational coefficients. It can be shown that, in general, one can find rational $u,v$ such that,
$$(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-...
- 205
5
votes
1
answer
462
views
Solutions of a general diophantine equation
So it turns out that there exist positive integers a, b, c and n, such that $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=n.$ See Estimating the size of solutions of a diophantine equation
Now I am ...
CommunityBot
- 1
-1
votes
1
answer
149
views
Find the diophantine-equations $3x(x^2+2)=y^2$ integer solution [closed]
Let $x,y$ be positive integers, such that
$$3x(x^2+2)=y^2$$
since
$$3\cdot 1(1^2+2)=3\times 3=9=3^2$$
$$3\cdot 2(2^2+2)=6\cdot 6=36=6^2$$
$$24\cdot 3(24^2+2)=72\cdot 578=204^2$$
so I have ...
- 2,075
12
votes
2
answers
1k
views
What is the rank of the Mordell equation $y^2 = x^3 - 2$?
The mordell equation $E$ defined by $y^2 = x^3 - 2$ over $\mathbb{Q}$ is known to have only one non-trivial integer solution $P = (3,5)$ from here. However, the rank of Mordell-Weil group $E(\mathbb{Q}...
- 2,607