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 ...
935 questions
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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 ...
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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 $NC$ ...
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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 $x^2+y^2+z^4+t^4$...
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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 $\...
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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 ...
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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 ...
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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 ...
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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.
...
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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 ...
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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 ...
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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
$$n(f, B)=\{(x_1,x_2)\in\mathbb Z^2| |x_1|\leq B, |x_2|\...
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Beyond pure rational and integral solutions to cubic equations
I started reading Silverman and Tate’s introductory book on elliptic curves. In the introductory chapter they mention that for the Bachet equation $x^2 - y^3 = c$, there are infinitely many rational ...
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The security of one-time digital signatures from a solution to a diophantine equations
I wonder how well arbitrary Diophantine equations can be used to make one time digital signature schemes.
For our one-time digital signature scheme, the public key is a collection of polynomials $f_1(...
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Is there a known connection between Wieferich primes and the Goormaghtigh conjecture?
I posted this question on SE, and was told I should repost it here.
The Goormaghtigh conjecture explores the Diophantine equation of the form
$$
\frac{a^b-1}{a-1}=\frac{c^d-1}{c-1},
$$
where $a>c&...
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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 ...
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Can $12n+5$ be written as $2x^2+5y^2+9z^2+xyz$ with $x,y,z$ nonnegative integers?
Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, for any $n\in\mathbb N$ we can write $4n+1$ as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$.
Motivated by this, here ...
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What are the integer solutions to $y^3=2x^3+x+1$?
The question is in the title.
Short motivation. Consider Diophantine equations in $2$ variables. Quadratic ones are easy, and can be solved, for example, here https://www.alpertron.com.ar/QUAD.HTM. ...
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Jones–Sato–Wada–Wiens diophantine equation [closed]
I came across this from the 1993 book Matiyasevic - Hilbert's 10th problem. Typeset from another question:
\begin{align}
P(a,b,\dotsc,z)=(k+2)\Bigl(1&-(wz+h+j-q)^2\\
&-\left[(gk+2g+k+1)...
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Integer points on genus 1 curves using CAS
How can I practically find integer points on genus 1 curves with small coefficients using computer algebra systems (CAS), like Mathematica, Maple, SageMath, Magma, etc.?
As a specific example, do ...
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Conjectures about the automorphism group of integer lattice by enlarging the matrix
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Notation: $\GL(n, \mathbb{Z})$ to be the set of $n \times n$ invertible matrix, and ...
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Diophantine equation about the automorphism group of lattice by constraints
Fixed $\sigma_x=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\right)$ and $K=\left(
\begin{array}{ccc}
3 & 32 & -64 \\
1 & 32 & -32 \\
-2 & -32 & 64 \\
\...
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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 ...
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On the equations $(x^n+1)(y^n+1)=z^2+1$ and $(x^n-1)(y^n-1)=z^2+1$
Note that
$$(1^2+1)(2^2+1)=10=3^2+1\ \ \mbox{and}\ \ (1^4+2^4)(5^4+6^4)=8^4+13^4.$$
Today I tried to find positive integers $x,y,z$ satisfying $(x^4+1)(y^4+1)=z^4+1$ but failed. In view of this ...
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Finding number fields over which Diophantine equations are solvable
Given a Diophantine equation $f(x_1, \dots, x_n) \in \mathbb{Z}[x_1, \dots, x_n]$ and a family of number fields $K$ (say, the number fields of a specified degree and signature), are there techniques ...
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How to solve special Diophantine equation systems (which one can solve by hand) with the computer?
I have a quadratic Diophantine equation system which is possibly not homogeneous and has some mixed and some linear terms.
But I know that there are only finitely many solutions over the integers.
One ...
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Cubic surface in $\mathbb{P}^3$ singular along a line
Maybe it is a stupid question but I'm not able to find the answer anywhere else.
My goal is to prove in an "algebraic geometry fashion" that $\sqrt{n}$ is not a rational number for $n$ not a ...
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Distribution of number of integer solutions in box to bivariate polynomials?
Take a bivariate polynomial of degree $d_x+d_y>\max(d_x,d_y)>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree).
What is the ...
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Integer solutions of Diophantine equation $y^2= 1+4n^{\underline k} $
I am looking for the integer solutions for the diophantine equation $y^2 =4n(n-1)(n-2)\cdots (n-k+1)+1$ for a given $k$ where $n>k+1>5$.
In other words,
$$y^2=1+4n^{\underline k},\tag{I}$$
where ...
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Diophantine system
Consider a sequence of integers $n_i,\ i=1,\ldots, N$ and $\nu_k=\sum_{i=k}^N n_i$. Consider a sequence $\Delta_i,\ i=0,\ldots, N+1$ with $\Delta_i\in \{0,1\}$ and $\Delta_0=\Delta_{N+1}=0$. For $i=0,\...
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Solutions of equation $\sin \pi x_1\sin \pi x_2=\sin \pi x_3\sin \pi x_4$ [closed]
I am interested in finding all the solution $(x_1,x_2,x_3,x_4)\in \mathbb{R}^4$ of equations:
$$\sin \pi x_1\sin\pi x_2=\sin \pi x_3\sin\pi x_4.$$
I have found out a paper: Rational products of sines ...
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Recursively obtained hard Diophantine equation for "Baseless numbers"
An equivalent problem was originally asked on MSE as Does every number base have at least one “Baseless number”?, but did not receive any answers that would help answer the main question about "...
- 611
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On a structural decomposition of polynomials based on integral roots
Given an irreducible polynomial of structure $$f(x,y)=\sum_{\substack{i,j\in\{0,1,2\}\\i+j\leq3}}a_{i, j}x^iy^j\in\mathbb Z[x,y]$$ with $a_{2,1}a_{1,2}a_{1,1}a_{1,0}a_{0,1}a_{0,0}\neq0$ if $f(m,n)=0$ ...
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$n$-variable polynomials modulo $p$
The Hasse-Weil bound implies that for any 2-variable polynomial $P(x,y)$, there exists approximately $p$ solutions in $\mathbb{F}_p$ of $P(x,y) \equiv a \pmod p$ for sufficiently large $p$, and any ...
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Write $p^2$ as $x^2+2y^2+3\times 2^z$ with $x,y,z$ nonnegative integers
In April 2018, I noted that the first integer $n>1$ with $n^2\not\in\{x^2+2y^2+3\times 2^z:\ x,y,z=0,1,2,\ldots\}$ is $$5884015571=7\times17\times49445509.$$
Question. Is it true that for each ...
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On Kellner's result and the Erdos-Moser equation
Let $m, k$ be positive integers. Consider the Erdos-Moser expression $S_{k}(m) = 1^k + 2^k + ... + (m-1)^k$. By a result of Kellner, we know that if $m | S_{k}(m)$, then $m|B_k$, where $B_r$ denotes ...
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On sparse $0/1$ linear equations solvable with compressed sensing
If you have a system of $m$ linearly independent equations in $n$ variables with domain $0/1$ and we know there is at least one solution with at most $d$ variables to be $1$ then if $m$ at least a ...
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How small can $u$ be in the Pell equation $u^2-k^3 v^2=\pm 1$?
Let $k$ be positive integer, not a square and let $u_k,v_k$ be non-trivial
solutions to the Pell equation $u_k^2-k^3 v_k^2=\pm 1$.
Q1 How small $u_k$ can be infinitely often as function $k$?
This ...
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On the equation involving Stirling numbers of the second kind ${n\brace a}{m\brace b}={k\brace c}$, and its solutions satisfying certain requirements
In this post we denote the Stirling numbers of the second kind as ${r\brace s}$ and we consider the proposal to ask if the equation of the title has infinitely many solutions $${n\brace a}{m\brace b}={...
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pari/gp "bnfisintnorm" as poor man (quadratic) Thue equations solver?
For simplicity explaining only the quadratic case.
Given integers $n,m$, pari/gp "bnfisintnorm" finds $X,Y$
such that $X^2+n Y^2=m$ working in the number field
with defining polynomial $x^2+...
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Explicit bivariate quadratic polynomials where Coppersmith is better than standard solver?
http://www.numbertheory.org/pdfs/general_quadratic_solution.pdf gives a general method to solve quadratic bivariate diophantine equation while Coppersmith introduced a method to solve bivariate ...
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Probability of small solutions to an uniform random linear diophantine equation?
Take the set $$T(c_1,\dots,c_t)=\{(x_1,\dots,x_t)\in\mathbb Z^t\backslash\{(0,\dots,0)\}:\sum_{i=1}^tc_ix_i\equiv0\bmod q\}$$ where $c_1,\dots,c_t\in(-q/2,q/2)\cap\mathbb Z$.
What is probability ...
- 13.9k
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On segments of the series $\sum_p\frac1{p-1}$
Here I ask a question concerning segments of the divergent series
$$\sum_p\frac1{p-1}=\sum_{k=1}^\infty\frac1{p_k-1},\tag{$*$}$$
where $p$ runs over all the primes, and $p_k$ denotes the $k$-th prime.
...
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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 ...
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On the diophantine equations $x^n+n=y^m$ and $x^n-n=y^m$
Here I ask a question concerning the diophantine equations
$$x^n+n=y^m \quad \ \text{with}\ m,n,x,y\in\{2,3,\ldots\},\tag{1}$$
and
$$x^n-n=y^m \quad \ \text{with}\ m,n,x,y\in\{2,3,\ldots\}.\tag{2}$$
...
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Does each integer have the form $x^4-y^3+z^2$ with $x,y,z$ positive integers?
Let $\mathbb Z^+$ denote the set of positive integers. Here I ask the following question.
QUESTION: Does each integer have the form $x^4-y^3+z^2$ with $x,y,z\in\mathbb Z^+$?
I guess that the answer ...
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A Bachet equation [closed]
How to prove the Bachet equation
$y^2=x^3+45$
has no solution in $\mathbb{Z}$?
This is an exercise in Elementary Number Theory and Its Application, section 13.2.
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Diophantine equation $z=(ax+by+c)/(dxy)$, references? [closed]
I am looking for some sources (books or papers) which discuss the Diophantine equation
$$
z=\frac{ax+by+c}{dxy}
$$
where $a,b,c,d$ are given positive integers. Could anyone give some references?
...
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Class number of the cyclotomic tower
Let ${\Bbb Q}(\zeta_{\infty})$ be the field obtained by adjoining all roots of unity. We define
Cl(${\Bbb Q}(\zeta_{\infty})$)$\colon= \underset{m > 1}{\varinjlim}~{\mathrm{Cl}}({\Bbb Z}[\zeta_m])...
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2nd order Diophantine Equation, when does it have solution(s)? [closed]
I have this problem:
Ax^2 + Cy^2 + Dx + Ey + F = 0, (B = 0 => Bxy)
I need to know under which circumstances the above has solution(s).
Thank you for your time.
- 11
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System of diophantine equations with restricted set of solutions [closed]
I'm engineer, not mathematician, so excuse me for wrong terminology, but I hope you'll understand the problem.
Example situation: I have N electronic components. Each of them has reactance and ...
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