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 ...
838
questions
2
votes
0
answers
66
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 ...
- 121
6
votes
1
answer
699
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?
- 153
0
votes
1
answer
95
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 ...
- 3,477
1
vote
0
answers
119
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 $x^2+y^2+z^4+t^4$...
- 23.8k
3
votes
0
answers
210
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 $...
- 23.8k
3
votes
0
answers
239
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: ...
- 23.8k
4
votes
3
answers
386
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 ...
- 8,452
0
votes
1
answer
60
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 ...
- 1,699
6
votes
1
answer
551
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 ...
- 72.5k
1
vote
0
answers
88
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 $\...
- 23.8k
2
votes
0
answers
78
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 $f$, ...
- 23.8k
2
votes
0
answers
89
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,...
- 23.8k
4
votes
0
answers
119
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 ...
- 3,477
8
votes
2
answers
484
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 ...
- 3,477
0
votes
1
answer
155
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
382
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 $d=1$, ...
- 3,477
5
votes
1
answer
386
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 ...
- 3,477
3
votes
0
answers
138
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 ...
- 6,770
1
vote
0
answers
206
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 ...
- 21
1
vote
1
answer
212
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 ...
- 83
-3
votes
2
answers
151
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 ...
- 741
0
votes
0
answers
71
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 ...
- 1
5
votes
1
answer
182
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 = ...
- 443
-1
votes
1
answer
90
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?
3
votes
2
answers
169
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 ...
- 443
1
vote
0
answers
138
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 ...
- 21
1
vote
1
answer
119
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}{...
- 121
1
vote
1
answer
231
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. ...
user178594
5
votes
1
answer
280
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?
- 13.2k
1
vote
0
answers
75
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.
...
- 11
1
vote
0
answers
58
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
679
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)(...
- 651
4
votes
2
answers
729
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$$
has infinitely many positive integral solutions if we take $c=a+b$ and $a^2+ab+b^2=d^2$ further assuming that $GCD(a,b,c)=1$.
Main problem: Find some ...
user178594
0
votes
0
answers
155
views
Representing integers as sums of three powers
A famous open question, discussed several times on MathOverFlow, asks Which integers can be expressed as a sum of three cubes in infinitely many ways?. This is open even for $n=3$, that is, we do not ...
- 3,477
1
vote
0
answers
106
views
Integral points in smooth cubic curves
Let $X$ be a smooth affine cubic curve in $\mathbb A^2$ defined by $f(T_1,T_2)\in\mathbb Z[T_1,T_2]$ (of course $\deg(f)=3$ by definition), and
$$n(f, B)=\{(x_1,x_2)\in\mathbb Z^2| |x_1|\leq B, |x_2|\...
- 393
1
vote
0
answers
68
views
Beyond pure rational and integral solutions to cubic equations
I started reading Silverman and Tate’s introductory book on elliptic curves. In the introductory chapter they mention that for the Bachet equation $x^2 - y^3 = c$, there are infinitely many rational ...
- 4,312
1
vote
1
answer
107
views
Solutions to diophantine equation related to an interpolation problem on hypercubes
Question:
which $n$ and $k$ satisfy $\frac{k^n-1}{2^n-1}\in\mathbb{N}$?
The motivation for the question is a constraint on the cardinality of interpolation-constraints for the $2^n$ corners of a ...
- 11.9k
2
votes
1
answer
185
views
Mod n, are all higher powers also lower powers?
Since there are only finitely many residues mod $n$, there is some function $H(n) \le n$ such that for all integers $n>1$, $r$, and $e>H(n)$, if $r$ is an $e$-th power mod $n$ then there is some ...
- 8,710
7
votes
1
answer
287
views
Rational points on regular curves over global fields
Let $k$ be a global field and $C$ a smooth projective curve over $k$ which is not isotrivial. Then there is a well-known trichotomy:
If $g(C) = 0$ and $C(k) \neq \emptyset$, then $C \cong \mathbb{P}^...
- 19.4k
1
vote
0
answers
122
views
The security of one-time digital signatures from a solution to a diophantine equations
I wonder how well arbitrary Diophantine equations can be used to make one time digital signature schemes.
For our one-time digital signature scheme, the public key is a collection of polynomials $f_1(...
- 25.5k
0
votes
0
answers
165
views
About Diophantine equations
I am a self studying student and I am interesting with diophantine equations. I have the following questions:
How can I know that a diophantine equation is solved or not?
the second question is what ...
- 17
2
votes
0
answers
81
views
Integers solutions of products of truncated Riemann zeta functions
Let $n \in \mathbb{N}$ be a positive integer.
It is possible to prove that the equation $F_1(n)=m$ where $m \in \mathbb{Z}$ and
$$
F_1(n)=(1+2+\ldots+n)\cdot\left(\frac{1}{1}+\frac{1}{2}+\dots+\frac{1}...
- 1,303
0
votes
0
answers
43
views
Does this approximate linear Diophantine Equation have bounded number of solutions?
Consider the linear diophantine equation
$$\alpha u+\beta v =r+ \delta$$
where $\alpha,\beta,r\in\mathbb Q$ are known and their binary expansion has $O(k)$ bits to exactly represent them and $\delta\...
- 13.2k
0
votes
1
answer
321
views
Good references to study Baker's theory
I am studying diophantine equations and I need the theory of Bakers, Can you advise me about good books, or lectures on Baker's theory?
- 17
0
votes
0
answers
42
views
Question in a proof of Diophantine Sets
I started to read and learn about The extension of Diophantine Sets. Now I'm reading this paper : The $D(1)$-extensions of $D(−1)$-triples $\{1, 2, c\}$ and integer points on the attached elliptic ...
2
votes
0
answers
339
views
Classifying solutions of a certain Diophantine Equation
The following question arose from a problem I am working on.
Let $N, k$ be positive integers. Consider the Diophantine equation in $a, b, c$:
$$
\frac{1}{a} + \frac{N - 1}{b} = \frac{N^k}{c}
$$
with ...
- 383
1
vote
2
answers
269
views
Integral solutions of quadratic equation $5 X² − 14 XY + 5 Y² = n$
Solve for all integers $x$ and $y$ the quadratic form $5 X² − 14 XY + 5 Y² = n$ for some integer n. I know that for some cases there are recurrence solutions, but I'm not sure how to solve these ...
- 741
1
vote
1
answer
140
views
Can $P(z)$ have a divisor in a given congruence class?
In the answer to this previous question , Noam D. Elkies proved that for any integer $x$, $x^3-x^2-2x+1$ can only have a divisors equal to $-1$, $0$, or $1$ modulo $7$. I would like to know what is ...
- 3,477
5
votes
0
answers
145
views
Can $z^3+1$ be powerful for integer $z$ other than $-1,0,2$ and $23$?
In 1976, Schinzel and Tijdeman proved that if a polynomial $P(z)$ with integer coefficients has at least $3$ simple zeros, then there may be at most finitely many $z$ such that $P(z)$ is a perfect ...
- 3,477
11
votes
1
answer
792
views
How to describe all integer solutions to $x^2+y^2=3z^2+1$?
The question is in the title. Here is a short motivation. The general quadratic Diophantine equation is
$$
x^TAx+bx+c=0,
$$
where $x$ is a vector of $n$ variables, $A$ is $n \times n$ matrix with ...
- 3,477