Questions tagged [diophantine-equations]
Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. Topics: Pell equations, quadratic forms, elliptic curves, abelian varieties, hyperelliptic curves, Thue equations, normic forms, K3 surfaces ...
901
questions
3
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2
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What can be said about the cube-free part of $x^3 -3xy^2 +y^3$?
For $x$ and $y$ in $\mathbf{Z}$, not both zero, let $cfp(x,y)$ be the cube-free part of $x^3 -3xy^2 + y^3$ (normalized to be $> 0$). One sees:
(#) $cfp(x,y)$ is either a product of primes $p$, with ...
1
vote
2
answers
611
views
Describe all integer/rational solutions to $x^3+y^3+z^3+t^3+s^3=0$
The question is in the title.
Equation $\sum_{i=1}^n x_i^3 = 0$ has no non-trivial integer solutions for $n=3$. For $n=4$, there are known descriptions of all integer/rational solutions, see
Choudhry, ...
6
votes
0
answers
284
views
A generalization of the Diophantine $m$-tuple problem
Are there distinct positive integers $a_1,a_2,a_3,b_1,b_2,b_3$ such that $a_ib_j+1$ is a perfect square number for each $i,j$ ($1\leq i,j\leq3$)?
I asked the following question in a group, and ...
4
votes
0
answers
259
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Equations involving sum of fourth powers
Do there exist rational numbers $x,y,z$ such that
$$
\quad \quad z^3 - 1 = x^4+y^4 \neq 0 \tag{$a$} \quad ?
$$
Also, do there exist rational numbers $x,y,z$ such that
$$
\quad \quad z^3 - z = x^4+y^4 \...
1
vote
0
answers
150
views
How difficult is to find rational points on these genus 3 curves:
How difficult is to find all rational points on these genus 3 curves:
$$
(a) \quad \quad x^3 + y^3 x +y^2 - y = 0
$$
$$
(b) \quad \quad x^4 - y^3 + x y + x = 0
$$
Short motivation. Consider the ...
1
vote
2
answers
124
views
Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$
Let $f, g \in \mathbb{Z}[x]$ be coprime polynomials.
I am interested in an upper bound for
$$
N(B) = \# \{ x \in [-B, B] \cap \mathbb{Z}: f(x)\mid g(x) \}.
$$
I assume there must be something known ...
0
votes
2
answers
216
views
$y^3=x^4+x$, and computing all rational points on rank $0$ Picard curves
What are the rational solutions to the equation
$$
y^3 = x^4 + x,
$$
in particular, are there any (finite) solutions other than $(x,y)=(0,0)$ and $(-1,0)$?
Context: This is the simplest-looking ...
0
votes
0
answers
75
views
Number of solution to homogeneous linear Diophantine equations
Let $T,M\in\mathbb{N}$ be fixed. Consider a linear Diophantine equation of the form
$a_1 x_1 + a_2 x_2 + … + a_n x_n = 0 $
with $a_i \in [-T,T] \subset \mathbb{Z}$. Is there an asymptotic formula to ...
3
votes
0
answers
499
views
Regarding the Challenge Problem in 3Blue1Brown's most recent video: Will $\binom{x}{4}+\binom{x}{2}+1=2^k$ for $x>10$? [duplicate]
Link to the video here with timestamp
In deriving the formula for regions of Moser's Circle Problem, it observed that the formula
$$
F(x)=\binom{x}{4}+\binom{x}{2}+1
$$
achieves values that are equal ...
5
votes
0
answers
168
views
Existence of large integer solution for a simple-looking equation
Is it true that for every $k>0$ Diophantine equation
$$
y^2 + x^2y + z^2x + 1 = 0
$$
has an integer solution $(x,y,z)$ such that $\min\{|x|,|y|,|z|\}\geq k$?
Motivation: this equation arises in the ...
1
vote
1
answer
185
views
Solvability of two-variable quadratic equations with a parameter
(a) Prove that there exist infinitely many values of integer parameter $a$ such that equation
$$
2 x^2+a x y+y^2+1 = 0
$$
is solvable in integers $(x,y)$.
(b) The same question for a similar equation
$...
2
votes
1
answer
292
views
On the equation $4y^p= x^2 + 3$
Do there exist some non-zero rational numbers $x, y$ such that $x \neq \pm y$ and
$$4y^p = x^2 + 3 \tag{1}$$
for some odd prime $p$?
If one lets $y=a/b$ and $x=c/d$ where $a, b, c, d$ are non-zero ...
2
votes
1
answer
250
views
An arithmetic problem involving a system of equations
Fix a positive integer $r$. Describe the solutions to the system of equations given by:
$$\begin{equation}\sum_{1\leq i\leq r}X_i^2\equiv0\pmod{X_k}(1\leq k\leq r)\end{equation}$$
Example: In the case ...
1
vote
0
answers
29
views
Does $2$ variable linear Diophantine equation in $NC$ imply $2$ dimensional shortest vector is in $NC$?
Consider the Linear Diophantine in known $a,b,c\in\mathbb Z$
$$ax+by=c.$$
Above can be solve by Extended Euclidean which is not in $NC$ as far as we know. It is clear if Extended Euclidean is in $NC$ ...
0
votes
0
answers
70
views
Lowest asymptotic bound to $4^n - 2v_n^2$ where $v$ is an odd integer, $n$ fixed
The general problem is this. I try to find a positive integer $\delta_n$ such $qv_n^2 +\delta_n = p\cdot 4^n$. More precisely, I am looking for a lower bound (depending on $n$) for $\delta_n$ as $n\...
10
votes
2
answers
1k
views
Integer solutions of an exponential equation
How can I solve this equation?
$$7^{x} +2=y^{2}$$
$x$ and $y$ must be natural numbers.
2
votes
0
answers
90
views
Persistence of KAM tori as a function of dimension
I have tried posting this question in MSE, but I think it might be too technical so I'm trying again here.
In KAM theory one tries to describe the persistence of quasi-periodic motion when an ...
4
votes
1
answer
834
views
Does this conic have a rational point?
Consider the conic
$$C = \{X^2+uY^2+vZ^2=0\}\subset\mathbb{P}^2_{\mathbb{Q}(u,v)}$$
over the function field $\mathbb{Q}(u,v)$.
Does $C$ have a $\mathbb{Q}(u,v)$-rational point?
1
vote
1
answer
197
views
Rational points on genus 3 curves defined by short equations
(a) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3-y=x^4-x.
$$
(b) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3+y=x^4+x.
$$
If not a complete answer, I would be happy to ...
1
vote
0
answers
128
views
Is every even number greater than $44$ not divisible by $8$ of the form $x^2+y^2+z^4+t^4$?
Related to this question,
where Bogdan Grechuk suggested this question.
Q1 Is every even number greater than $44$ not divisible by $8$ of the form $x^2+y^2+z^4+t^4$...
3
votes
0
answers
232
views
Is $16a+5$ always of the form $x^2+y^2+z^4$?
Working over the integers.
For $a$ up to $10^7$, $16a+5$ is always of the form $x^2+y^2+z^4$.
Q1 Is $16a+5$ always of the form $x^2+y^2+z^4$?
Heuristic argument:
For prime $p=4b+1$, both of $p$ and $...
3
votes
0
answers
254
views
Are all odd integers greater than $599$ of the form $x^2+y^2+z^4+t^4$?
For $a \le 10^7$, the equation over integers $4a+1=x^2+y^2+z^4+t^4$
has solutions.
Q1 Is it true that all integers of the form $4a+1$
are also of the form $x^2+y^2+z^4+t^4$?
Heuristic argument: ...
7
votes
3
answers
559
views
Question on a crucial lemma in Euler's approach to Fermat's Last Theorem for $n=3$
As many of you may know, the illustrious L. Euler put forward a proof of the case $n=3$ of Fermat's Last Theorem via infinite descent. The thing is that, at a certain point, he resorted to the ...
0
votes
1
answer
105
views
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 ...
6
votes
1
answer
626
views
Hilbert's tenth problem for equations with finitely many solutions
Is there a known example of a set $S$ of Diophantine equations such that
$S$ is computable;
it is a theorem that every equation in $S$ has (at most) finitely many solutions;
the function that maps an ...
1
vote
0
answers
97
views
Hardness of solving $0=\sum_{i=1}^k \operatorname{linear}_i(x_1,\ldots,x_n)^D$ over the rationals
This is related to cryptography and this question
and another question.
In short, we are asking about decomposing multivariate polynomial
as sum of perfect powers of linear polynomials.
Working over $\...
2
votes
0
answers
86
views
Complexity of finding solutions of trapdoored polynomial?
Related to this question Cryptography signature scheme based on hardness of finding points on varieties.
Working over $K=\mathbb{Q}[x_1,...,x_n,y_1,...y_m]$.
By abuse of notation, for polynomial $f$, ...
2
votes
0
answers
94
views
Cryptography signature scheme based on hardness of finding points on varieties?
Related to this question Complexity of finding solutions of trapdoored polynomial.
I am trying to build signature scheme based on hardness
of finding points on varieties.
Let $K$ be field and $M=K[x_1,...
4
votes
0
answers
126
views
Rational points on a cubic surface with small coefficients
Do there exists integers $(x,y,z,t)\neq (0,0,0,0)$ such that
$$
2x^3+2y^3+z^3+t^3+2x^2y-2z^2x-y^2z-z^2t = 0 ?
$$
A short motivation: there are many known counterexamples to the Hasse principle for ...
8
votes
2
answers
575
views
Existence of rational points on a generalized Fermat quartic
Question: Do there exist integers $(x,y,z)\neq (0,0,0)$ such that
$$
13x^4+11y^4=8z^4 ?
$$
Some motivation: This is currently the smallest (in a sense defined here On the smallest open Diophantine ...
0
votes
1
answer
167
views
How can we solve the following number theory problem? [closed]
Let $m$ and $n$ be positive integers less than $2000$ which satisfies the equation $(m^2-mn-n^2)^2=1$. How can we determine the largest possible value of the expression $m^2+n^2$?
2
votes
2
answers
405
views
Existence of rational points on generalized Fermat quintics
Do there exist integers $(x,y,z)\neq (0,0,0)$ such that
$$
(a) \quad 2x^5+3y^5=6z^5
$$
$$
(b) \quad x^5+3y^5=7z^5
$$
Here is a short motivation. Equation $ax^d + by^d=cz^d$ is trivial for $d=1$, ...
5
votes
1
answer
440
views
Existence of rational points on some genus 3 curves
Do there exist a pair of rational numbers $(x,y)$ such that
$$
(a) \quad x^4+x^3+y^4+y-1=0
$$
$$
(b) \quad x^4+x^3+y^4+y^2-1=0
$$
Magma function IsLocallySoluble returns that both equations are ...
0
votes
0
answers
151
views
A question on the Hilbert-Kamke problem
The Hilbert-Kamke problem consists in studying the integral solutions of the Diophantine system
$$
x_1^i + \dots + x_s^i = n_i \text{ for } 1\leq i\leq k
$$
with $x_i\geq 0$ for $i = 1,\dots,k$.
I am ...
1
vote
0
answers
312
views
How would one go about solving this conjecture concerning exponential Diophantine equations?
I’ve been working on the Collatz Conjecture, and I believe I’ve reduced it to a more tractable problem. Unless there are some errors I’ve overlooked, I have managed to reduce the Collatz Conjecture to ...
1
vote
1
answer
256
views
On the Diophantine equation $a^3 + b^3 = c^3 + d^3$
Let $a, b, c$ and $d$ be positive integers. What are the conditions that $a, b, c$ and $d$ should satisfy for the equality $$a^3 + b^3 = c^3 + d^3$$ to hold? In particular, can $a, b, c$ and $d$ be ...
-3
votes
2
answers
160
views
Non-vanishing of this ternary quadratic expression [closed]
I'm dealing with the expression $x^2+y^2+6z^2+8xy+4x+4y−6xz−6yz$. I want to show that this expression is always non-zero whenever $x,y$ and $z$ are positive integers. How does one do this? (Note that ...
1
vote
0
answers
158
views
On quadratic Diophantine equations with n variables
Consider the following problem. Given a quadratic equation
$$ \sum_{i,j=1}^n a_{i,j} x_ix_j + \sum_{k=1}^n d_{k} x_k + e = 0, \qquad a_{i,j},d_k,e\in\mathbb{Z}$$
if it exists, find (at least) a ...
2
votes
1
answer
235
views
System of two linear Diophantine equations
Let $n\in\mathbb{N}$ be a positive integer. Denote by $f(n)$ the number of integral solutions of the following system
$$
\left\lbrace\begin{array}{l}
\sum_{i=1}^nx_i = 3n; \\
\sum_{i=1}^n (2i-1)x_i = ...
-1
votes
1
answer
94
views
Diophantine equation $546\cdot p+546\cdot q=1001\cdot r$ [closed]
$546\cdot p+546\cdot q=1001\cdot r$
$p,q$ odd primes, r positive integer.
are there infinitely many solutions?
And what if r is a Catalan number?
1
vote
2
answers
228
views
Solutions of a linear diophantine equation
Let $N(h)$ be the number of solutions of the following linear diophantine equation:
\begin{equation}
x_1 + 2x_2 + 3x_3 + \dots + (h-1)x_{h-1} = 6h-6;
\end{equation}
where $h\geq 2$ and solution means ...
1
vote
0
answers
146
views
Exponential diophantine equation that I’m curious about
For which $x,y \in \mathbb{N} $ does the following hold? $\forall k \in \mathbb{N} \exists a,b,c,d \in \mathbb{N} \cup \{0\} : x^{a} + x^{b} = y^{c} + k y^{d} $. What sort of restrictions do we need ...
1
vote
1
answer
162
views
Solutions to some cubic Diophantine equations
In searching for integral points on elliptic curves, I am encountering Diophantine equations of the following forms:
$3m^3 - n^3 = {2^a}{3^b}$, $4m^3 - n^3 = {2^a}{3^b}{5^c}$, $5m^3 - n^3 = {2^a}{3^b}{...
1
vote
1
answer
246
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. ...
5
votes
1
answer
299
views
Parity of number of solutions to Diophantine equations
By $MRDP$ resolution of Hilbert's tenth, we infer, counting number of solutions to Diophantine equations is undecidable.
Is parity of number of solutions to Diophantine equations undecidable?
1
vote
0
answers
120
views
When does a system of homogeneous quadratic equations have integer solutions?
I learned that in general, solving systems of quadratic Diophantine equations is a difficult problem. But I wonder if there are special (and non-trivial) types of systems that are easier to handle.
...
1
vote
0
answers
61
views
Set from a diophantine equation with similar statistics to primes
While doing some computational calculations with some diophantine equations, I came across with some sequences from solutions of quartic and quintic equations with slowly decreasing frequency, similar ...
7
votes
2
answers
723
views
Integer solutions of an algebraic equation
I'm trying to find integer solutions $(a,b,c)$ of the following algebraic equation with additional conditions $b>a>0$, $c>0$.
$(-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2) + 2 a b (-a^2+b^2+c^2)(...
4
votes
2
answers
740
views
On the equation $a^4+b^4+c^4=2d^4$ in coprime positive integers $a\lt b\lt c$ such that $a+b\ne c$
Background: The equation
$$a^4+b^4+c^4=2d^4$$
has infinitely many positive integral solutions if we take $c=a+b$ and $a^2+ab+b^2=d^2$ further assuming that $GCD(a,b,c)=1$.
Main problem: Find some ...
0
votes
0
answers
172
views
Representing integers as sums of three powers
A famous open question, discussed several times on MathOverFlow, asks Which integers can be expressed as a sum of three cubes in infinitely many ways?. This is open even for $n=3$, that is, we do not ...